The Eastern half of the empire was still Roman.
But everyone calls it Byzantine and it was culturally different... particularly Byzantine "Rome" of 1000 AD v. Rome of 1 AD
Even though Byzantium didn't conquer and wipe out the Romans.... So the name changed and the culture changed even the language of the majority of the population ended up different... But the people called themselves Romans and the Empire never exactly fell in the transition
So Transition to a "different civ" could mean...
You were wiped out... want to try again with something new? (and a few memories of those you defeated)
OR
You survived and changed
The more the player can control things which make EITHER of those interpretations make sense, the more fun the game will be.
So a Rome Civ could switch to Mongols, Change their civ name to Mongols and rename all of their cities from the Mongol list.
The Roman empire was razed to the ground and the Mongols resettled it for the good pasture land (like Genghis's initial plans for China...) (those towns are filled with mongol settlers... a few straggling survivors build the old way)
OR
A Rome Civ switched to the Mongols, kept the name Rome and all new cities are named from the Roman list.
Roman Generals becoming more powerful to the point of leading warbands across the empire during the Crisis.. breakthroughs in cavalry archery technology and tactics (some copied from invaders), allowed a particular commander to unite the warring generals and lead the empire into the next great phase of conquest.
Either of those stories are a Rome->Mongol Civ switch
If the Player can Easily ie by default control
1. The civs name
2. The city list options
Then the player can go for the feel/history they want.
The default for AIs (and probably human players) should probably be switch to new, but with portions of the old... but the option to keep all of the old (particularly in names) would be important.
This could be in
civ name
city name
city graphics
"flags"
(That way Rome and Aztecs and even Poland can into space)