There can be only one pope, if more than one person claim to be the pope he must consider that the other claimants are anti-popes. What is possible is that different regions get patriarchs that are almost fully independent in administrative tasks and everyday practices, which probably is enough to appease most of the rulers. But they need to be in full communion with the doctrine and tradition of Rome. You can even make the pope less powerfull giving to ecumenic councils the ultimate power about dogmatic questions, but if the church is not universal it is not catholic.
I was picturing them, at least initially, having very few differences between themselves and the Pope in Rome besides their location and the specific politically motivated changes that lead the ruler to decide to declare an Anti Pope. When Protestantism began until around about the Treaty of Westphalia, Catholics saw Protestants as a threat to Catholicism's existence in a way you just described.
Also, Ecumenic Councils could be formed to handle doctrine issues within a nation and open up a national dialogue on where religion should go within the nation. This would probably come about as more religious and social issues are brought up in the years to decades after the setting up of their Anti Pope.
I suppose you could theoretically have a situation where every (major) country has its own Antipope, each claiming that he's the one true Pope and all the rest are just pretenders, but this seems pretty unlikely and unstable (since having so many claimants to the Papacy would be a sign that something had gone very wrong, there'd be huge pressure to find a way to solve it and find one Papal candidate whom everybody can agree on).
Well it certainly wouldn't happen over night. Over the course of years, Anti Popes being set up would result in a lot of instability, but I could see new Popes in Rome becoming increasingly entrenched in seeing these national Anti Popes as the enemy and somebody to stand firmly against, entrenching the position of the Anti Popes.
I think a war, or series of wars, akin to the Thirty Years War will happen with a resulting treaty (or treaties) with a similar effect as the Treaty of Westphalia, recognizing that there are all sorts of different Papal leaders in different states and that everybody's just going to have to live with these different beliefs.
Well, there is the example of Henry VIII, who was the first in a long line of "national Popes" in the form of the English monarchy. Similarly, the Lutherans and the other "magisterial" protestants tended to have some hierarchy. I think the question of church lands means that even without Luther, a number of monarchs are going to be very curious about some change in doctrine.
That's another thing, this is more or less the monarchs seizing power from the church by making a church leader who does not hold international sway, but only power within their country.
What???
First, Protestantism was the religion of the upper crust, wealthy burghers, nobility and monarchs. Reading the bible in the vernacular is irrelevant if one can't read or afford an expensive book.
The veneration of saints was also a popular thing. Contrary to learned theologians, the common people considered heaven to mirror the social structure they lived in. You just didn't appeal to the king or emperor if you had grievances, you went to the local lord or city councillor. Mirroring that, you prayed for the intercession of a saint, not directly to god. Abolishing the veneration of saints, while it had solid theological reasons, was mostly done to grab all those gold and jewel-encrusted shrines and to shut down those channels the pesky commoners had to reach God. It wasn't popular at all.
Neither was the dissolution of the monasteries. Those provided a social safety net for the poor. Slashing that to piss away the loot for parties and palaces was basically what killed the Wirtschaftswunder of the late middle Ages and started the rapid decline of the commoners' rights and powers in the modern Age (I'm talking about actual power, like the public's right to demand that the authorities pardon the convicted at a court Hearing, not entirely theoretical and toothless stuff like 'popular sovereignity').
This is very interesting, do you think monasteries and saints could have maintained a level of importance?
Also, I could see more radical national Catholic Churches breaking away from saints and seizing the wealth of corrupt religious centers within their country.
I'm just curious, the popes accrued to themselves quite a bit of political power. Some, during the Reformation called the pope more a king than a priest. Now, if you have a bunch of different antipopes, said gentlemen (maybe women, since in folklore there was a Pope Joan) are going to be claiming that political authority in whichever nation they get set up in. The pope was considered by some to even outrank the emperor (the emperors denied this and it got ugly OTL), and had the power to make or break kings at their will. Which means that (for instance, France) is going to be divided between what the king decides in Paris, and the "pope" decides in Avignon (?). Both are going to be claiming to have power over the other (king trumps pope, since pope is a French subject; pope trumps king since all kings are answerable to God), which is possibly going to lead (IMHO) to an increase in the commonality of there being Henry II-St. Thomas Becket-type conflicts... Basically, you're turning a monarchy into a diarchy since there are now two kings (for all intents and purposes) claiming to be God's anointed. Could get messy...
I don't see how this would arise. This would be assuming that the King and the Anti Pope were the only two powers (which would be ignoring the nobility, powerful lords, the merchant class, and, to a lesser extent until later, the people). And this would be assuming that the Anti Pope would be a power separate from the King, even though the King appointed him to his role.
I think what would happen is the Anti Pope be the second most powerful person in the country but, it would still be subservient to the King in most instances. If the King appoints the Anti Pope and the Anti Pope's successors, while also acting as the nation's leader, than the King is unquestionably the one in charge. What would be interesting are the little powers the Anti Pope would have over the King. I could see traditions developing where the King must be crowned by the Anti Pope and I doubt the King could just fire the existing one (so if a king dies, his successor would have to just live with whoever the current Anti Pope is until that guy dies and he can appoint his Anti Pope).
I don't see how this would be a diarchy. The Anti Popes are celibate and would not have an heir outside of who the King appoints or maybe who they or some sort of religious council pick.
The Orthodox consider themselves just as universal as Catholics.
This.
It's exactly the reason why I don't see the issues people are having about it being unstable and politically delegitimizing to the Catholic Church, because Protestantism was the same way. After decades of warfare and political conflict, I can't see how this breakdown wouldn't be a permanent situation, with tensions dying down.