Under what circumstances can it be legitimate to have parallel popes? I am think for example of a senario in which some part of the world were to be cut off from the rest of the world. How would Catholics in that part of the world fare? At first everything could go on as normal, but at some point it would become necessary to appoint new bishops or rule on questions of faith. Would they be allowed to appoint from their own number a sort of parallel pope who would be vested with those powers until such time as the isolation could be overcome and connections to the outside world resumed.
I am thinking in particular of the Celtic Church. After the fall of the Roman Empire, connections to Rome were more or less severed and as Paganism returned to most of Europe, Catholicism continued in Ireland and and missionaries from the Celtic Church eventually carried their faith to there countries. Finally, the Celtic Church was reunited with the Roman Church at the Synod of Whitby, in which the Celtic leaders effectively recognized the Pope’s higher authority. Prior to that, did the Celtic Church have some sort of Pope of their own? Or how did it arrange its internal affairs, appoint bishops etc.
It is by all means desirable that a bishop should be appointed by all the bishops of the province. But if this is difficult because of some pressing necessity or the length of the journey involved, let at least three come together and perform the ordination, but only after the absent bishops have taken part in the vote and given their written consent. But in each province the right of confirming the proceedings belongs to the metropolitan bishop.
The ancient customs of Egypt, Libya and Pentapolis shall be maintained, according to which the bishop of Alexandria has authority over all these places since a similar custom exists with reference to the bishop of Rome. Similarly in Antioch and the other provinces the prerogatives of the churches are to be preserved. In general the following principle is evident: if anyone is made bishop without the consent of the metropolitan, this great synod determines that such a one shall not be a bishop. If however two or three by reason of personal rivalry dissent from the common vote of all, provided it is reasonable and in accordance with the church’s canon, the vote of the majority shall prevail.
The point is that the Metropolitan in those days had the ability to confirm the appointment of bishops. If we were, somehow, to return to communion with the Orthodox, I would imagine that the appointment of bishops would likely return to this type of mold (where the Pope would be responsible for the election of bishops within the Latin Church, the Patriarch of Constantinople for the Greek Orthodox, the Patriarch of Alexandria for the Copts, etc.)
As for matters of doctrine where there were disputes between individual bishops in a province, that is what local councils were for (though keeping in mind that a local council could not define the Universal Magisterium and could be overruled by an Ecumenical Council or the Pope if they went off the rails)
There have been areas which have been physically separated from Rome for some centuries. Isolated bishops can consecrate priests and their own successor bishops, if necessary. There is little need to “rule on questions of faith.” The faith does not change. They just need to continue to practice the same Catholic Apostolic faith and the same seven sacraments. The Catholics of Ireland, India, Lebanon did a wonderful job of doing exactly that in past centuries.
I would be careful about referencing the Celtic church. Much of it is really the imagination of early Anglican writers who like to think of a church holidng out against the over-bearing power of Rome (a bit similar to how the legends of King Arthur were invented to give the English a history of resistance against foreigners). In truth, the ‘Celtic church’ (most historians today prefer the term ‘Insular Christianity’) were fully part of the Latin Church, albeit with some Irish and British traditions not found elsewhere. The Council of Whitby did not ‘reunite’ two separate churches but settled the way the date of Easter was to be calculated (there were many variations then, and even until now).
Were there churches whch were isolated from the rest of the world? There were a few examples but in all but one case the isolation was not so complete.
Ethiopians - they were unknown to the West (Crusaders searched for Prester John, for which the Ethiopians were a likely candidate) but they had their own head, the Abuna, who was in communion with the Coptic Patriarch of Alexandria
Malabarese of India - they were ‘discovered’ by the Portugese when they arrived in India at the end of the 15th century. They have their own bishops and was at that time in communion with the Assyrian Patriarch of Babylon
Maronites of Lebanon - they were ‘discovered’ by the Crusaders. In truth, they fled to the Lebanon mountains to escape Christological disputes and later, Muslim invasion. They were supported by the Popes in the Christological disputes and by the Byzantine emperor against the Muslims. They were not forgotten by the Christian world, only by the Roman church.
Hidden Christians of Japan - these were the only instance of Christians cut off from the rest of the world I can think of. After the institution of the Shoganate in the 17th century, Christians were persecuted - clergy were expelled and those who remained as well as the laity were killed and some crucified. Japan then shut its doors to the outside world until the arrival of the US Commodore Perry in 1854. Behind those shut doors, the hidden Christians of Japan continued to baptise their children and maintained the faith, even without priests. Their prayers were designed to sound like Buddist chants but still contained some untranslated Portugese and Latin words. The liturgy and the Bible were passed down orally, as possesion of Christian books were punisable by death. The story was that - when Japan reopened its doors and a French priest built the first Catholic church in Nagasaki, a Japanese farmer family wandered into the church one day before it opened and looked at all the statues with a look on their faces like they have come home and then went up to the French priest to ask which among the statues was Mother Mary.
The story of the Hidden Christians of Japan teaches us that it is possible to preserve the faith for centuries even without any popes, bishops or priests. God will find his own.