In the last part of his life Tertullian aligned himself with the New Prophecy, a movement that had begun in Phrygia around Montanus and two women, Prisca and Maximilla, and that claimed continuing prophetic utterance, expected the end shortly and demanded a harder discipline: no second marriage, no flight in persecution, longer fasts. His later works argue exactly those positions and speak of the psychici, the merely soulish, meaning the mainstream church.
What that alignment amounted to institutionally is not clear. Augustine, writing two centuries later in the same province, says there was a separate body called the Tertullianists which returned to the Catholic church in his own day. Modern scholarship is divided about whether that is reliable, and about whether Tertullian ever formally left the church at Carthage at all, since the New Prophecy in Africa may have operated as a rigorist current inside it rather than beside it.
The consequence for his reputation was permanent. He is quoted constantly and he is not a saint, and the later tradition handled him by taking the vocabulary and leaving the man. Jerome lists him among the illustrious writers and repeats a report that he was a presbyter, which the works themselves nowhere confirm.