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Your challenge gents and ladyfolk is
Without resurrecting the Byzantine Empire from nothing (so no ERE because England says so) have the English Kings and Byzantine Emperors co-operate so that either:
a) The English and the Greek are equally great members of a Anglo-Greco-Roman Empire (choose whom absorbs whom)
b) The Roman Empire/Byzantium and the English/British Empires are two of the pre-eminent powers of the day.
Bonus Points : PoD before 1204 (or part of the 4th Crusade). End the Great Schism. English Patriarch of similar standing to Rome or Constantinople.