In my most interesting game of Europa Universalis II to date, I played Byzantium. The game started in 1420, with Constantinople only ruling Thrace and the Morea. I took advantage of the Frank-ruled Duchy of Athens's war with Albania to wage a peaceful conquest of the former Greek-populated state, then forged a military alliance with a dynamic Empire of Trebizond that managed to conquer its immediate hinterland in northeastern Anatolia. A decade of peace followed, during which I built up my armies and engaged in productive diplomacy with the courts of Orthodox Europe and even with certain of the more notable Frankish states. And then, I unleashed my armies on an Ottoman Empire distracted by wars with other Turkish states in Anatolia, liberating first Rumelia then the eastern shores of the Aegean and Marmara. I managed to restore the eastern empire of Rome, even as Trebizond became Europe's march and the Greek cities of the Crimea declared their independence in a miniature empire stretching into southern Ukraine. If not for an attack by the Venetians, fearing Byzantine interests in their Hellenic island possessions, Byzantium would have been nicely poised to take part in Europe's 15th and 16th century expansion. Already, I already had trade colonies on the Arabian coast.
If ever I write alternate-history fiction, I'd like to include this as a fragment of background. What effect might a revived and dynamic Byzantium have had on the world?