Why are my AC vents sweating?

A supply register is cold. When humid air reaches it, the water in that air condenses on it, and you get sweating or dripping.

Sometimes the cause is as simple as where the bath fan is. A bath fan is there to remove moisture, and if it’s placed where it can’t catch the shower moisture, that humid air crosses the room and hits a cold register. Finding where the moisture is coming from is the job of a diagnostic.