So it seems the Constitution allows it, but it would be unprecedented (on the same evidence).No, they can't.
"It would obviously be oppressive and unfair if a prosecutor, disappointed with an acquittal, could secure a retrial of the accused person on the same evidence, perhaps before what the prosecutor ‘considered to be a more perspicacious jury or tougher judge’."
What can happen, Bats, is for the prosecutor to appeal the sentence.