I'm sure Mike R has checked out from this discussion long ago, but I'm interested in his comments on the following.
I was intentionally stretching the scenario to illustrate a point. And please stick with me here... I'm not trying to be a troll that refuses to listen to reason or read my rulebook! (If I was, I'd go post on the baseball board!)
Now, take my scenario, and have the ball in the air over foul ground, but F3 camped under it is now in the running lane... and also change it so that BR's interference is not malicious or flagrant... (which is, as Dakota mentions, the only rule that supports disallowing the run ... ps - thanks Dakota for being the fish on that bait, I needed someone to say that!
)
So... a ball that WOULD HAVE BEEN foul (however you wish to describe it) had it been either caught or allowed to land can score a run if and only if interference is ruled on the batter and the runner has crossed the plate before interference occurred.
Therefore ... if there's no interference and the ball is caught, R1 is a dead duck and doesn't score. If there's no interference and the ball lands and stays foul, R1 goes back to third. ... But if the OFFENSE commits an illegal act as described above, the OFFENSE benefits. (Note that this is why my early questions about "Was the ball caught before the INT" and "Was the ball fair or foul" were important... at least to me.)
This is so wrong.
(PS - The initial question implies, but doesn't confirm, a catch. A ball being bobbled can also be "knocked loose" as posted in the initial question)
PPS - So sorry it took so long for me to get to my point!