According to Wikipedia,
indigenous Christianity in Algeria after the Arab conquest is supposed to have declined due mostly to the lack of the Christian monasticism that allowed the Copts to persist. Most Catholic monasticism is under the Rule of St. Benedict, who lived in 480-547 CE, but his "rule" was not codified until much later.
Augustine was of course the Bishop of
Hippo west of Carthage. He wrote several letters on monasticism to the community there, and
spread monasticism in neighboring communities to Carthage.
It seems like a major blow to Christian monasticism and Trinitarian Christianity in this region came even earlier than the Arab conquest, due to conflict with the Arian Vandals.
Honeric soon came into conflict with the Catholic bishops, trying to seize their property and banish them to Corsia &c. in favor of Arian bishops. The banished Catholics apparently took Augustine's body to Sardinia. Honeric only ruled for seven years, the worst persecution happened in the year of his death, and he was apparently universally despised. Perhaps if his nephew
Gunthamund had succeeded him a few years earlier... It seems Catholicism began to revive under
Hilderic, but he was deposed and murdered, which provoked the Byzantine re-conquest of Tunisia. Christianity increased under Byzantine rule, and it seems numerous Christians fled to Carthage from the Arab conquest.
History of Roman-era Tunisia
History of early Islamic Tunisia
In general, it only seems like there are reports of local surviving Christian communities when a Christian power seizes control of the region. Wikipedia
reports that the medieval Christian community that survived into the
Kingdom of Africa lived in
Gafsa. But it also seems like Berber was the dominant prestige language until the 13th century or so.
I don't mean to push the Christianity so strongly, but this has been an interesting way to read up on some history I don't know much about. For a region on the periphery of so many empires, it seems extremely cosmopolitan. There might be some interesting influence from
Ladino later on.
EDIT
That translated passage was quoted by the last Vanal king,
Gelimer.