I feel like a world in which the Arabs push into Anatolia becomes a different world. To this day, the Roman Republic - I know "Empire" is what the Kontostavros bureaucracy likes to say, but theoretically we've had
demokratia since the
Epanastasi - is really associated with Fortress Asia Minor. In all honesty that was a close call, historically: The defeat of the Arabs and the subsequent fall of Persia saw the old Empire become so complacent that all it would've taken after that was for the Karluks to turn west and attack the Muslims instead of invading the Indus. They'd have smashed right through Arabia and found the eastern border of Anatolia poorly-defended. The renewed border wars with the Hashimid Caliphate did, at least, lead to the eastern frontier becoming pretty formidable after that period.
Probably a loss of Anatolia results in Greek as a language being confined to Hellas. It might even result in a more compact empire which can better defend its western half, or keep the Roman Catholics out. The Magyars displacing the Bulgars north of the Danube was bad enough, but their conversion to Latin Christianity and the bickering between Rome and Constantinople over the matter really curtailed the ability of the Empire to project power west of Constantinople and north of the river.
re-established the Western Empire
For a few years, anyway. The marriage of the Lombard Queen Brunechildis to Emperor Eustathios I certainly had a lot of potential and did bring Italy back into the fold for the better part of a century, but her ability to exercise power in Gaul was pretty terrible, and those areas kind of splintered for good after the Easter Schism.