[jcranmer]

Yes, there's a rupture between Eastern and Western Christianity, but Western Christianity still accepts the authority of the pope to speak on behalf of Christianity, and there's still a sense that they're still part of the same Christendom, just disputing who is going to come out on top. You might compare it to the modern "One China Policy" in that both Taiwan and China see themselves as the legitimate government representing all of China. Note that the Holy Roman Emperor and the Byzantine Emperor both titled themselves as Emperor of the Romans--they're still claiming heir to the same unified Christendom.

(And also note that the latest Byzantine Emperors repeatedly tried to mend the schism to secure Western aid in stabilizing their empire.)