In C.E. 647, The Byzantine Exarch of Africa (known as Gregory the Patrician) declared independence from Byzantium. His motivations were both religious and political, dealing with the debate between Monotheletism and Chalcedonianism and the failure of Constantinople to stop the Arab invasion of Egypt.
However, Gregory died before his ambitions could be realized. At the battle of Sufetula in 647/8, he was slain in a conflict with the caliphate, who seem to have had 40,000 troops during the battle. Information on Gregory's troop numbers has proved elusive, save for a citation-less claim somewhere on Wikipedia that he had 100,000 Amazigh men during the battle, which seems dubious at best.
A more detailed description of the battle says that originally, the Arabs sent a force of 20,000 soldiers, which resulted in a pitched battle with Gregory's forces. However, Ibn Al-Zubayr arrived with an additional 20,000, and they proceeded to be victorious.
After the battle, the Muslims retreated from Africa on the agreement that Gregory's successor pay them a hefty annual tribute, being unable to storm the Byzantine fortresses the defenders retreated to. Byzantine rule in Africa was severely weakened by Gregory's rebellion, although the new Exarch continued providing the correct tax to Constantinople.
So, enough of the background! What would be most likely to happen if Gregory had not died in battle, and as a result would have won the battle? Or the other way around, depending on when Al-Zubayr's forces get there. Would we see an independent North African state, or would it just slow the Muslims down a bit? Interested to hear any opinions on the matter!