If a creature with persist dies and returns with a -1/-1 counter, does undying work the next time it dies?

Persist and undying are two separate triggered abilities with different conditions. Persist returns a creature from the graveyard if it had no -1/-1 counters on it when it died (CR 702.79a), placing a -1/-1 counter on it when it returns. Undying returns a creature if it had no +1/+1 counters on it when it died (CR 702.92a), placing a +1/+1 counter on it when it returns.

These conditions are checked independently. The presence of a -1/-1 counter (from persist) does not affect undying's condition, which only cares about +1/+1 counters. So if a creature has both persist and undying, each ability can trigger on a different death, provided its specific counter condition is met.

Importantly, when a creature returns with a -1/-1 counter (from persist) and a +1/+1 counter would be on it (from undying), the two counters annihilate each other per CR 121.3, leaving the creature with neither counter type — potentially resetting both abilities for future deaths.

Concrete example: You control a Cauldron of Souls-enchanted 2/2 with undying (giving it persist). It dies with no counters — persist triggers and returns it with a -1/-1 counter, making it a 1/1. It dies again with no +1/+1 counters, so undying triggers and returns it with a +1/+1 counter. The -1/-1 and +1/+1 counters cancel out (CR 121.3), leaving a clean 2/2 with no counters — ready for either ability to trigger again.

The key rule: each ability checks only for its own counter type at the moment the creature dies. They do not interfere with each other's conditions (CR 702.79a, 702.92a).