In my limited experience, one small tub for 5 months, it seems there are two kinds of foam. The type of foam that comes from the foaming agents in soap from suits and from body lotions will tend to persist after the agitation stops. This type can be reduced by oxidation with chlorine, bromine or MPS. The type of foam that forms while the jets and air injectors are running but dissipates almost immediately after the agitation is stopped probably comes from other causes, and does not seem to respond to oxidation. If I were guessing, I would say it is from the various types of dissolved solids that build up over time.
I'm still on first fill after 5 months, so the dissolved solids are starting to build up. We have foaming while the air is running that can get 3 or 4 inches deep in the center of the tub, but it dissipates within about a minute after the air is stopped. Even taking the sanitizer to shock level doesn't seem to have much effect on this. We've just learned to live with it. It isn't much of a problem really, as it doesn't interfere with our enjoyment. We usually just run the air bubbles for 5 minutes or less of our soak anyway, if we use the air at all.
Incidentally, I started keeping the bromine level between 6-10 ppm after having a soap type foaming incident. Before that, I was targeting 4-6 ppm. The higher level seems to help keep the water clearer and the foaming minimal.