What can be done so that modern Christianity is incompatible with the nation state, and instead had something like a caliphate in the past?
What would be the implications?
Considering his Christianity spread to China, Africa, the Caucaus, and Southern India before it really got to Western Europe, and still retained some presence for the next two thousand years, I would say that you won't be managing to get them all under one ruler, whereas Islam allowed that as they were group moving mostly through conquest, at least at the start. Also the issue how Caliphs and Sultans tied their positions together and often claimed to be descended form Muhammad or one of his followers. And Arabic was spread as being the only thing to read the Quran or do government in, to a greater degree than Latin, which was only in Western Europe anyways. So really, I don't see any way for this to happen. From the very start the Disciples were to go to the ends of the earth speaking different languages to different groups, with political power having no real role in the New Testament outside of showing the corruption, decadence, or heartlessness of dynastic leaders.