USA and Ukraine citizenship, how to travel with two pports when duel citizenship isn't allowed by Ukraine?

My wife is a Ukrainian and recently became an American. Ukraine doesn't allow duel citizenship. Does anyone know how to travel between the two countries without having to renounce citizenship of either. She can enter Ukraine on her US pport but can only stay for 90 days (we are there much longer). If she enters on her Ukrainian pport, she has to have a US visa in it upon travel to the States. So, we are kind of in a bind. Do we just us the US pport or is there a way around?