How could the Pope have remain in Avignon once the Schism ends rather than returning to Rome? what would be the far reaching effects?
You'd need a successful Muslim invasion if you're talking something medieval.
If you want to go for something more off the wall and 19th century a successful Socialist or Communist revolution in Italy that's strong enough to stay in the saddle could do it and France offers Avignon as a renewed Papal residence.
Considering the legacy of those days and the legacy of the French Revolution and such, maybe the Pope would go to another place than Avignon... Like Spain or Portugal by example.
Oh, Byzantium would let him stay, no question. WOULD he stay? THAT's the question.And even for Byzantines...
The opposition to the status of Pope by Orthodoxes is a bit exagerated - from what I got, the Orthodoxes refuse the 'primacy' of the Bishop-Patriarch of Rome, but they consider the position legit per see, I think.
Itally in the hands of the Emperor of Byzance, the Pope's powers would be surely quite reduced, but they would let him there. Why not? He is one of the Old Grand Patriarches.