What are real ties to Rome?
Some sort of tie in to the Roman state apparatus.
Things that claimed the title due to conquest ir whatever don't really count.
So no Sultanate of Rum, Holy Roman Empire Moscow as Third Rome etc.
I think there's multiple dayes one could claim personally I go with Byzantine as it was a political continuation of Rome, were Roman citizens and saw themselves as Roman.
The Byzantine fragments are somewhat plausible as a state while the last Romans are somewhat plausible as late as the 19th century when inhabitants of the last areas of the Byzantine state still saw themselves as Romans over Hellenes.