During the times of Pope and Anti-Pope, which one had the Keys of Peter? The French Pope or the Italian one?
Whichever Pope was legally appointed or elected. The Pope is Bishop of Rome regardless of where he happens to live at a particular moment.
You seem to be under the impression that the Avignon Popes were antipopes. There was no pope in Rome for all that time. Then, one day, the cardinals who were in Rome, decided to elect a pope when a pope had already been elected in Avignon. So you see that the schism started, not in Avignon, but in Rome.