And as the first poster wrote, the ball becomes dead on the infraction for the T, so illegal contact during the screen is ignored by rule and not a foul.

Right, but if these happen at the same time wouldn't that be a false double, or are you saying the T trumps the common foul?

you could have a personal followed by double Ts that would offset and resume POI -- so only the penalty for the personal would be administered. And that wouldn't be a throw-in at the division line (unless that's where the foul was, of course)

A foul during the administration of the FTs for the T?

If that were to happen, somebody (or more than one somebody) is getting tossed. And we'll probably shoot a few more with the lane cleared.

If you wanna see a foul during a FT turn into a disaster look up the end of the Charlotte/Richmond A10 tourney game last year.

http://www.youtube.com/watch?v=3XUl2y43Fkg

Sure, but it would be a personal foul (intentional or flagrant) which is not necessarily resumed with a TI at the division line.

A number of situations could fit, by rule.

1: B1 steals the ball from A1 and streaks down the court for a layup, wide open. A1 screams at you to "call the damned foul!" As per the case play, you hold the whistle on the T until B1's shot is in the air. B1 is fouled on the shot.
(I'll be honest, I'm not sure I'd rule this one that way, but this seems to fit.)

2: A1 begins his shooting motion when B1 runs his mouth at the nearest official. A1 goes airborne, releases his shot, and comes down on B2 who had a) or b) had not established LGP in time. In a, you have a false double foul. A will shoot 2 FTs for the T, and B will get the ball OOB at the spot nearest the PC foul. In b, it's a false multiple foul. A will shoot two FTs for the T, then 1 or 2 FTs for B2's foul with the lane occupied.

3: During the FTs for the technical foul, B1 goes and stands directly behind the shooter, but within his legal area. A2 takes exception and goes up and grabs B1 and pulls him backwards. By rule, this is an intentional (maybe even flagrant) personal foul, and B will get the ball at the spot nearest the foul following B1's FTs.

I'm not suggesting it's something you'd see very often, or that it would be something that should necessarily be called. I was just trying to supply a possible answer for the question. To be honest, I wasn't even sure it was a correct answer, thus the question mark at the end.