As far as I'm aware the Supreme Court has not specifically considered the issue of whether inconsistent theories of prosecution in two different trials violates due process. The federal appellate courts have been pretty negative on the practice. See 653 F.3d 426 (6th Cir. 2011).[1]

That being said, criminal trials are inherently statistical and that admits some level of inconsistency. If the standard of conviction is say 95% certainty, the evidence might prove a 95% probability of guilt, while still leaving a range of facts undefined (e.g. where exactly was Halbach killed?) In fact, it's possible for the evidence to establish a 95% probability of guilt for two different people, even when only one could be guilty.

Thus, the focus of courts reviewing these sorts of cases seems to be not on whether the theories are inconsistent, but on the prosecutor's ethical duty to submit only evidence they believe to be true. In the Stumpf case, the prosecutor presented in Wesley's trial a statement from a jail-house informant that Wesley admitted to being the one to pull the trigger. He did not present that evidence at Stumpf's plea hearing, and in a subsequent proceeding with Stumpf, argued that the jail-house witness was not credible. The problem was not that the two theories were inconsistent, but that in either the trial or the later hearing, the prosecutor must have advanced evidence he thought to be false.

[1] In that case, there were two defendants accused of participating in the murder, but it was unclear who actually "pulled the trigger." One, Stumpf, pled guilty to aggravated murder, and in his plea hearing the state argued he was the primary shooter. He was sentenced to death. In the trial of the other, Wesley, the state argued that he was the primary shooter. He was sentenced to life. Stumpf filed a habeas petition in federal court. The Supreme Court held that the inconsistent theories did not require vacating Stumpf's guilty plea, because under Ohio law an accomplice who did not actually pull the trigger could be guilty of aggravated murder. However, it remanded to the Sixth Circuit to determine whether the inconsistent theories necessitated re-sentencing. On remand, the Sixth Circuit held that the inconsistent theories did violate Stumpf's due process rights and granted his habeas petition, unless Ohio granted him a re-sentencing.