The Eastern Roman reconquest of North Africa in the 530s AD ended the Vandal kingdom there. But trade declined until the Islamic invasions of North Africa in the 600s AD and the establishment of the Umayyad, and then the Fatimid dynasties.
Under Islamic rule, North Africa became part of a large trading network again. Carthage was abandoned in favor of the new Islamic cities of Tunis and Mahdia. At the same time, improvements in camel caravans made it possible to maintain trade routes across the Sahara to Ghana and Mali, which became important for the gold, salt, and slave trades.
By the 1100s, North Africa was independent again under the Almohad dynasty, and then about 1200 AD it broke up into even smaller kingdoms. These corresponded roughly to the modern countries, under the Hafsids in the east (modern Libya and Tunisia), the ‘Abd al-Wadids in the middle (modern Algeria), and the Marinids in the west (modern Morocco).
All of these countries traded a lot with Italian cities like Genoa and Pisa and Venice. By the 1500s, however, the growing Ottoman Empire conquered these kingdoms, and pretty much all of North Africa was part of the Ottoman Empire.
Umm El Madayan: An Islamic City Through the Ages
by Abderrahman Ayoub, Jamila Binous, Abderrazak Gragueb (1994)
The Late Roman West and the Vandals by Frank M. Clover (not a kids’ book) (1993)