This is an oddly phrased question. It seems to assume that there was some clear understanding in the east of an absolute necessity of being in communion with Rome.
… what was your average eastern layman to do when he realized his church leaders were no longer in communion with Rome? Move? Thanks.
Essentially they came to realize that Rome was no longer Orthodox.
The relationships between the patriarchates were much more like what we might call ‘strategic partnerships’ (for lack of a better term). They were all trying to accomplish the same thing, the salvation of souls, and worked together for that goal … each in his own regional sphere of influence. It was not a case of the Pope and his staff (in those days) actually administering the eastern churches (naming bishops, writing canons, proclaiming saints or declaring dogmas), as might be true today for Catholics. The local Patriarch with his synod pretty much did everything that needed such attention.
Under the canonical rule prevalent in those days, there was one bishop per each city. In large cities (like Constantinople) there were inevitably foreigners, and there were parishes of the ‘Latin Rite’ there, but under the Patriarch of that city. Likewise, there were some parishes where the ‘Greek Rite’ was served (such as in the city of Rome) under the Pope.
In those times when communion was impaired between the major patriarchs, all the parishes of whatever ritual practice would follow their own local bishop.
The same thing happened during the Great Western Schism, when there were two or three Popes. The local parishes followed their local bishops, who usually (in those days) followed whichever Pope that their king supported.
In my answer to the question, I think the average layperson just kept on doing whatever they usually did: kept working at their jobs and going to the local parish for their sacramental life. Usually only pilgrims, merchants and mercenaries had the kind of mobility that would make it possible to leave for another city, so perhaps some few did, in both directions.
However, for those unique individuals who may have had a personal conviction that said they either agreed with the Pope, or agreed with the Patriarchs, they (at first) would have had to physically relocate in order to remain in communion with the bishops of their choice. It could have happened.
During the Crusades (not very long after the schism seemed to really break), the Latin western church had named new Patriarchs and bishops for the Sees that they could physically occupy with troops. They had hoped to unify Christendom in this way, by (in effect) giving Latin bishops to the Greek Catholics (Orthodox).
The idea of replacing bishops ultimately didn’t work (the Orthodox church went ‘into the bush’ as they say), but the legacy of that effort is that the Latin church now had established a second overlapping hierarchy in the east, where there was none before. At this point the typical Orthodox Christian was looking at the Latin church as a separate church in their own home towns. The Orthodox east did not begin to organize a separate overlapping hierarchy in the west until much later, when Orthodox Christians began to migrate into western Europe in significant numbers, but that must have implanted the same idea in the heads of westerners.
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