North Africa was split off from the rest of the western empire in the 430s by the Vandals, a branch of the Roman army that went rogue and decided not to work within the increasingly ineffective western roman administrative system. The Vandals were technically barbarians (they were foederati - barbarians recruited to serve as a unit in the Roman army), but they came from the frontier regions which had undergone centuries of cultural blending between the Roman army and its Germanic neighbors, and they adapted to life in roman north africa very easily. Despite the negative connotations we associate with their name, recent archaeological evidence suggests the Vandals were actually very capable administrators. North Africa's economy continued to prosper, and its important trade goods (grain, wine, olive oil, and a type of pottery you find all over the mediterranean, african red slipware) continued to be shipped all over the mediterranean. The Vandals blended in so perfectly with Roman society that we wouldn't really know North Africa had been conquered if we didn't have textual sources describing the conquest.

Things were business as usual under the Vandals, until Justinian (the emperor of the eastern Roman empire) decided to reconquer it in the mid 6th century. There's more archaeological evidence for this conquest - it was very violent, and lots of infrastructure was destroyed. The Romans rebuilt a lot of it, and held north africa for about a century before being conquered by the Arab armies (which, incidentally, had recruited heavily from Roman barbarian soldiers - foederati - on the empire's eastern frontier, the Ghassanids).