How might Protestantism fare?
Perhaps the rival popes shelter each other's dissidents?
Depending on how much influence these dissidents have in their new homes, there might eventually be doctrinal divergences among the three churches.
Plus Protestantism, split up as it was, often became rather statist--Luther did PR for the German princes, Calvin ran his own little Iran in Geneva, and the Anglican Church supported the English monarchy (one cleric said that Elizabeth, by forcibly imposing religious uniformity, was only doing as Moses did).
A split-up Catholic Church, particularly one in a strong state like France, might be more vulnerable to the exertions of kings and we might have Avignonese-Catholicism becoming a French state church much like Anglicanism was in England. And in a weaker state, you might have the church dominating the government.