OPINION
BEA, Circuit Judge:We are asked to decide whether a federal habeas court can order a state court to re-sentence a defendant under a lesser-included offense that was not considered by the jury.
Gordon Andrew Douglas was convicted by a California jury of one count of first-degree murder in violation of California Penal Code § 187, and a separate count of arson of an inhabited structure in violation of California Penal Code § 451(b). The evidence at trial established that Douglas and his brother broke into Jack Clark’s home to commit a robbery. During the course of the robbery, Douglas stabbed Clark to death. Several hours later after Clark was killed, Douglas and his brother returned to the scene of their crime and set fire to Clark’s house to destroy any fingerprints they might have left behind. Douglas was sentenced to twenty-five years to life in prison on the murder count, with a consecutive sentence of eight years on the arson count. In April 1997, after Douglas’s direct appeals and state habeas claims were exhausted, he filed a habeas petition in federal district court.
In February 2008, the district court granted Douglas habeas relief on a single ground: there was insufficient evidence to support Douglas’s conviction for arson of an inhabited structure in violation of California Penal Code § 451(b). The state has not appealed this ruling, and we make no ruling regarding its validity. The only issue before us is the relief afforded. The district court vacated Douglas’s conviction under § 451(b), but remanded the case to the state court with instructions for the state court to enter judgment against Douglas for violation of § 451(c), arson of a structure.
On appeal, Douglas contends that the district court exceeded its habeas powers when it directed the state court to enter a judgment against him for a violation of § 451(c). Douglas further contends that imposition of a sentence under § 451(c) would expose him to double jeopardy in violation of the Fifth Amendment to the United States Constitution, incorporated *504as a limitation against the states by the Fourteenth Amendment’s Due Process Clause. Benton v. Maryland, 395 U.S. 784, 794, 89 S.Ct. 2056, 23 L.Ed.2d 707 (1969).
Indeed the district court did exceed its habeas powers when it directed the state to modify Douglas’s sentence. However, the Double Jeopardy Clause would not be implicated if the state court, of its own accord, were to re-sentence Douglas under § 451(c). We vacate the district court’s order, and remand with instructions that Douglas be granted a conditional writ of habeas corpus as to the arson count only.
I. The District Court’s Habeas Jurisdiction
Douglas was convicted of arson of an inhabited structure under § 451(b). The district court, relying on People v. Ramos, 52 Cal.App.4th 300, 60 Cal.Rptr.2d 523, 525 (1997), held Clark’s house was not “inhabited” under California law, because the sole resident was killed by Douglas before he set the fire.1 California did not cross-appeal this ruling, so we are constrained to proceed under the district court’s interpretation of California law. See Greenlaw v. United States, 554 U.S. 237, 128 S.Ct. 2559, 2564, 171 L.Ed.2d 399 (2008) (under an “unwritten but longstanding rule, an appellate court may not alter a judgment to benefit a non-appealing party”); id. (“it takes a cross-appeal to justify a remedy in favor of an appellee.”). Under the district court’s interpretation, Clark should have been charged with arson of a structure under § 451(c) instead of arson of an inhabited structure under § 451(b). Section 451(c) is a lesser-included offense of § 451(b).2
Nonetheless, the district court exceeded its habeas jurisdiction when it instructed the state court to revise its judgment to enter a sentence for arson of a structure. The power of a federal habeas court “lies to enforce the right of personal liberty.” Fay v. Noia, 372 U.S. 391, 430, 83 S.Ct. 822, 9 L.Ed.2d 837 (1963), overruled on other grounds by Wainwright v. Sykes, 433 U.S. 72, 87, 97 S.Ct. 2497, 53 L.Ed.2d 594 (1977). As such, a habeas court “has the power to release” a prisoner, but “has no other power[.]” Id. at 431, 83 S.Ct. 822. “[I]t cannot revise the state court judgment; it can act only on the body of the petitioner.” Id.
Here, the district court impermissibly attempted to revise the state court judgment when it ordered the state to resentence Douglas under § 451(c). The district court’s power under habeas corpus was either immediately to vacate the prisoner’s arson sentence, or to postpone such relief for a reasonable period to allow the state court properly to sentence the prisoner.3 See, e.g., Hilton v. Braunskill, 481 U.S. 770, 775, 107 S.Ct. 2113, 95 L.Ed.2d 724 (1987).
*505Under California law, a trial court can modify a jury verdict when the trial evidence establishes that the defendant is not guilty of the charged crime, but is guilty of a lesser included offense. Cal.Penal Code § 1181(6).4 The only difference between an arson conviction under § 451(b) and a conviction under § 451(c) is whether the burned structure was inhabited. The district court found that although there was insufficient evidence of inhabitation, the jury necessarily found all the facts essential to convict Douglas for a violation of § 451(c) when it found him guilty of a violation of § 451(b).5 Hence, a California state trial court can modify the judgment from a conviction under § 451(b) to a conviction under § 451(c). See People v. Jones, 199 Cal.App.3d 543, 549-50, 245 CaLRptr. 85 (Cal.App.1988) (modifying a judgment of conviction for arson of an inhabited structure under § 451(b) to arson of an uninhabited structure under § 451(c) where there was insufficient evidence that the building Jones burned down was inhabited). Because the state has the power to correct the constitutional error of a criminal conviction based on insufficient evidence,6 it should be given the opportunity to do so. See Chioino v. Kernan, 581 F.3d 1182, 1186 (9th Cir.2009) (“[Hjabeas remedies should not unnecessarily infringe on competing interests such as a state’s interest in the administration of criminal justice.”) (internal quotation marks and citation omitted).
Instead of directing the trial court to enter a judgment under § 451(c), the district court should have granted a conditional writ of .habeas corpus and ordered that Douglas’s conviction under § 451(b) be vacated only if the state court did not re-sentence him within a reasonable time, such as 90 days. The state court would thus have an opportunity to correct its own constitutional error.
II. The Double Jeopardy Clause
Douglas contends that were he re-sentenced under § 451(c), he would be subject to double jeopardy in violation of the Fifth Amendment to the United States Constitution, incorporated as a limitation against the states by the Fourteenth Amendment’s Due Process Clause. Benton, 395 U.S. at 794, 89 S.Ct. 2056. This contention is without merit. In general, if a conviction is overturned on appeal on sufficiency of evidence grounds, “[t]he Double Jeopardy Clause forbids a second trial.” Burks v. United States, 437 U.S. 1, 11, 98 S.Ct. 2141, 57 L.Ed.2d 1 (1978). To permit a second trial after a conviction is overturned on appeal would “afford[ ] the prosecution another opportunity to supply evidence which it failed to muster in the first proceeding.” Id. Here, though, the government presented sufficient evidence in the trial proceeding to convict Douglas of a violation of § 451(c), and the jury *506found Douglas guilty of all the necessary elements of § 451(c). In these circumstances, the Double Jeopardy Clause is not offended by entry of judgment on the lesser-included offense. All of the essential elements of § 451(c), a lesser-included offense of § 451(b), were necessarily found by the jury based on Douglas’s initial prosecution.
Although Douglas’s jury was not instructed on the lesser-included offense, judgment may still be entered on a charge of violation of § 451(c), because Douglas was convicted, not acquitted, of the greater offense. In United States v. Gooday, 714 F.2d 80 (9th Cir.1983), this court held “[i]f no instructions are given on lesser included offenses ... an acquittal on [a greater] charge necessarily implies an acquittal on all lesser offenses included within that charge. An acquittal on the explicit charge therefore bars subsequent indictment on the implicit lesser included offenses.” Id. at 82 (emphasis added) (citations omitted). The Gooday rule makes sense. If a jury is not separately instructed on a lesser included offense, an acquittal on the greater offense may imply acquittal on one or more of the elements necessary for a conviction under the lesser offense. In such a case, the jury might have found proof lacking of essential elements contained in both the greater and lesser charges. If the jury did acquit based on a finding of lack of proof of elements of the lesser-included offense, any subsequent indictment on the lesser offense would indeed violate the Double Jeopardy Clause. But, unlike in Gooday, Douglas was convicted of the greater offense. On the way to finding him guilty of arson of an inhabited structure, the jury had necessarily found adequate proof that Douglas committed arson of a structure. Thus, whether the jury was instructed on the lesser included offense is irrelevant.
Contrary to the dissent’s contention, our decision does not grant the prosecution “another bite at the apple.” In this circumstance, the prosecution necessarily already proved the elements of a lesser-included offense. Thus, our decision simply permits the state court to carve a smaller slice from a bite which the prosecution has already taken.
Nor does United States v. Vasquez-Chan, (9th Cir.1992), hold that the Double Jeopardy Clause is implicated by a trial court’s failure to instruct the jury on a lesser-included charge. Vasquez-Chan was convicted in federal court of conspiracy to possess, with the intent to distribute, five grams of cocaine. Id. at 548. On direct appeal from the federal conviction, we held the evidence was insufficient to support the conviction. Id. at 553. The government contended the court was permitted to enter judgment on the lesser offense of misprision of a felony.7 Id. In response, we laid out three “conditions necessary to the entry of such a judgment.” Id. at 554. One condition was “the jury must have been expressly instructed that it could find the defendant guilty of the lesser-included offense and must have been properly instructed on the elements of that offense.” Id.
But Vasquez-Chan does not control this case. Vasquez-Chan arose on direct appeal, not on an appeal from a habeas petition. Id. at 549. We reviewed Vasquez-Chan in the exercise of this court’s supervisory power over district courts’ procedures. 978 F.2d 546, see 28 U.S.C. § 1291. Because Vasquez-Chan did not arise on an appeal from a denial of a writ of habeas corpus, we had no occasion to consider *507whether a conditional writ should issue. Moreover, the claim of possible violation of the Double Jeopardy Clause never arose in Vasquez-Chan. Indeed, the Double Jeopardy Clause was not raised as a ground of appeal, nor was the Double Jeopardy Clause mentioned in the opinion. Vasquez-Chan thus spoke only to the question of whether and in what circumstances — as a matter of federal procedural law — a circuit court can modify a conviction from a greater offense to a lesser included offense. Hence, Vasquez-Chan does not prevent the California state court from entering a conviction for Douglas under § 451(c) under a conditional writ of habeas corpus.8
The district court’s order is VACATED. We REMAND Douglas’s habeas petition to the district court, and instruct the district court to issue a conditional writ of habeas corpus if the state court does not re-sentence Douglas within 90 days. The California state court may, in its discretion, enter a conviction for Douglas under § 451(c).
VACATED and REMANDED.
. We note, however, two distinctions between this case and Ramos. First, Ramos was a burglary case, not an arson case. 52 Cal. App.4th 300, 60 Cal.Rptr.2d 523, 524. Second — and more importantly — the already-deceased occupant in Ramos died of natural causes. Id. But here, Douglas murdered the inhabitant. Again, we proceed under the district court's interpretation of California law. The extension of Ramos to these facts strikes us as problematic, but not problematic enough to affect the ground for decision.
. Compare Cal.Penal Code § 451(b) (“Arson that causes an inhabited structure or inhabited property to burn is a felony punishable by imprisonment in the state prison for three, five, or eight years.”) -with Cal.Penal Code § 451(c) ("Arson of a structure or forest land is a felony punishable by imprisonment in the state prison for two, four, or six years.”).
. Of course, Douglas would not have been eligible to be released from prison in any event, because he had not yet served his sentence for the murder conviction.
. California Penal Code § 1181(6) provides, "[w]hen the verdict or finding is contrary to law or evidence, but if the evidence shows the defendant to be not guilty of the degree of the crime of which he was convicted, but guilty of a lesser degree thereof, or of a lesser crime included therein, the court may modify the verdict, finding or judgment accordingly without granting or ordering a new trial, and this power shall extend to any court to which the cause may be appealed[J” A California court can also modify a judgment of conviction to a lesser included offense when that judgment is collaterally attacked in habeas proceedings. See In re Bower, 38 Cal.3d 865, 215 Cal.Rptr. 267, 700 P.2d 1269, 1278 (1985).
. The jury was not instructed on the lesser included offense, but the elements are exactly the same except the burned structure need not have been inhabited at the time of the arson.
. See Jackson v. Virginia, 443 U.S. 307, 324, 99 S.Ct. 2781, 61 L.Ed.2d 560 (1979).
. Misprision of a felony criminalizes a person’s failure to report knowledge of a felony to the appropriate civil or military authorities. 18 U.S.C. § 4.
. The dissent contends that the "procedural distinction” between this case and Vasquez-Chan is "irrelevant,” because the "pertinent holding in Vasquez-Chan rests on federal constitutional grounds.” This conclusion is inaccurate. As noted, Vasquez-Chan did not mention the Double Jeopardy clause even once, and we reject the dissent's conclusion that Vasquez-Chan "implicitly” imputed the Double Jeopardy Clause because it cited two cases (in a footnote) which involved double jeopardy. But even if Vasquez-Chan implicitly imported a double jeopardy analysis, the two cases cited in Vasquez-Chan (and in the dissent) are distinguishable from this one. The first case cited in Vasquez-Chan is Gooday, which we have already distinguished. Supra at pp. 505-07. The rule in Gooday is that an acquittal on a greater charge bars subsequent indictment on a lesser-included offense. Gooday, 714 F.2d at 82. The second case, Forsberg v. United States, 351 F.2d 242, 247-48 (9th Cir. 1965), surveys cases which held that a conviction on a lesser-included charge implies an acquittal on the greater charge. Neither Gooday nor Forsberg implicates the issue here: whether a court may enter a conviction on a lesser charge when the jury necessarily found the defendant guilty of the lesser charge on its way to finding him guilty of the greater charge.