illustration (attribution, if any possible, is at the end of the article)
Five early schools might be grouped under the pudgala-vada denomination, two of which were of sizeable influence: the vatsiprutriya and the sammitiya.
Although the pudgalavada would now be considered as a "clearly erroneous interpretation" of the teachings of the Buddha, this might be in hindsight. The pudgalavada schools were successful and important in India until the extinction of Buddhism in India. They had not however disseminated beyond (e.g. towards China) and thus died when Buddhism died in India.
The fundamental starting point of pudgalavada is the rejection of two "levels" of truths in Buddhism.
While they could accept that some teachings had their meanings "drawn out" and other teachings not (i.e. should be interpreted), they were adamant that the various truths were of equal legitimacy or importance. The Theravāda tradition would theoretically agree with this, but "(conventional) truths in ultimate terms" easily become "ultimate truths" nonetheless…
One of the consequences of such an approach was that the teachings on karma and rebirth were seen as equally important as the teachings on the five aggregates and the anatta doctrine. It then remained to explain how the various teachings were compatible!
When the Theravāda ended with conclusion such as « deeds are found, but no doer », the pudgalavada would meanwhile ask « who is experiencing the fruits of karma? »
These pudgala-vada schools thus considered that a pudgala (or "person") existed, that ensures the karmic continuity.
The pudgala is ineffable: it may be discerned from the five aggregates but cannot be separated from them, it is neither the five aggregates nor different from them.
At the naive level, the pudgala might appear in flagrant contradiction with the anatta doctrine, but of course the supporters of these schools took all the precautions necessary for this not to be the case. By refusing to separate it from the five aggregates for example, by refusing to characterise the pudgala "in itself", they avoided to introduce contradictions.
While the pudgala provides some support for a form of permanency, it is not postulated as a permanent self! Like with some reciprocal action-reaction between envelope and content, the pudgala gives some "individuality" to a specific group of five aggregates, but also it is given "individuality" by them… thus neither is truly permanent or independent but the interaction results in a constantly-maintained individuality.
The contention with other schools is that they would accept "persons" purely as conventions or denominations (just like the "chariot"), while the pudgalavada asserted its "real" existence, affirming that the five aggregates alone wouldn't allow for karma to reach the person deserving the fruit.
The pudgala could also provide answers to « how does memory function, if all dharmas are momentary? » or « what is self-awareness, if all dharmas are momentary? », questions which were hard enough to explain the split between Theravāda and e.g. Sarvāstivāda.
It is to be noted that the Theravāda developed much of its own doctrine by opposition to the pudgalava schools, notably its very notion of "real" dharmas.
"Real ultimate existence" was thus defined in Theravāda, not as "independent existence" but as "existence without background", i.e. "existence which cannot be analysed / decomposed / reduced further". With this sort of definition, "real dharmas" are opposed to "composites", and a person is a composite of five aggregates, thus "not ultimately real".
One could argue that the Theravāda defined dharmas in a way specifically intended to reject the "real" existence of pudgala, in what they believe best captured the anatta doctrine.
The first Points of Controversy addressed in the Kathavatthu (part of the Theravāda Abhidhamma) are a refutation of the pudgala doctrine.
• Origins of the Abhidharma/Abhidhamma
• Five aggregates without a self
• How does karma "catch up" with the "right person"?