Nothing at all. Any AOO should only be possible if responding to the opportunity can be done without the responder being otherwise threatened. Basically, if you are otherwise melee engaged and threatened yourself when an opportunity presents itself then no go.
It would certainly cut down on the ridiculous chessboard gridlock effect. If you see a way through 4 opponents and all are already engaged then you should be able to move through without a problem. If you have to run by 3 guys just standing at the ready its a different story.
Well, we may be able to work out a simple patch. Like I said, the AoO mechanic itself is fine, it's the conditions that trigger it.
I posit that 2 fighters of equal level as opponents are pretty well occupied. If one gets distracted, it is the other's advantage: this could trigger an AoO
If it was a 2 on 3 fight of equal fighters, the solo fighter is in trouble. If one of the duo teams needed to quaf a potion, odds are good the soloist would be unable to take advantage of this because the other fighter is more then able to consume all his attention as in a 1 on 1 fight.
Thus, barring a multi-opponent feat or special ability or higher level of ability (a 20th level soloist vs. a 1st level duo would not be so challenged, and in fact would be waiting for such an opportunity to strike).
Presently, 3.x D&D treats those situations all the same. it lets the math work out the problem. Namely that a 20th level dude is likely not to get hit by 1st level dudes, even IF they get AoO.
An Equal level 2 on 1 has only the same moderate chance to hit if they get an AoO, so it tends to wash out. Keep in mind, that the soloist is facing double the number of attacks. IF one of the duo pauses to quaff, the threat he is facing is lessened already (because there's one attacker swingingg at him for the round).
I'd be OK with rules like:
you can only use an AoO if they are distracted (ex. quaffing) when facing one opponent.
Then make a Feat that increases that allowance by one (allowing for taking it multiple times).
Conceptually, by default, all fighters lack this feat. If you're facing an enemy, and somebody else runs through or teams up and quaffs, you get the benefit of them not hitting you, but not the free attack because you're not skilled enough to deal with multiple opponents more efficiently.