Yeah, I do wonder if it's possible to give just enough information that it looks like you have a 50% probability of committing the other burglary -- enough to prevent reasonable doubt that he's guilty of the first burglary, but not enough to say beyond reasonable doubt he's guilty of the second. But I don't think it actually works.

It's not quite the same, but I did once try to figure out what happened if you were charged with two related crimes, and they could prove you were guilty of one or the other but not both. And there was an obscure case where someone was in fact found guilty of the lesser of them. But I don't know if there's an official answer to what's supposed to happen.