Correctable error
 

Team fouls are 6 - 5, and at the next foul the official lines everybody up to shoot 1&1. The first shot misses, is rebounded and put back. As the other team takes the ball out of bounds for the throw in, it's brought to the official's attention that the fouls are now 6 - 6, and there shouldn't have been any free throws.

Obviously this is awarding an unmerited free throw. And it's within the correction time frame: "In order to correct any of the officials' errors listed in Article 1, such error must be recognized by an official during the first dead ball after the clock has properly started."

The question is, how do you resume. By 2-10-5 "Points scored, consumed time and additional activity, which may occur prior to the recognition of an error, shall not be nullified."

The miss and putback, to my mind, constitute "points socred, consumed time, and additional activity." So it would be proper to let that play stand. If there was to be a correction applied, it would be that the shooting team would be given the ball OOB for a throw in at the spot nearest the foul. You can't do that and give the other team the POI throw in along the baseline.

What is the correct thing to do here?

Let's say that A1 fouled B1 and B1 shot the FTs inappropriately, after which B2 scores the basket off the rebound.

Since there were no points scored on FT shots and no other "activity" during the FT to cancel, there's no correction to apply. (If B1 had made one or both FTs, then we'd "correct" the error by wiping those points. If there had been a personal foul during the FT, we'd wipe that too.)

The basket counts, since as you rightly point out, it occurs after the FT is completed.

Play resumes at PoI, which is a throw-in for team A at any point along the endline (that is, A can run the endline). This is different than what would have happened if no error had occurred, where B would have been given a throw-in at the spot nearest where the foul occurred.

By rule, it's a correctable error. To correct it, you do, nothing?
POI is B ball now. You can't undo the action.

BITS,
The way I am reading the play is that Team B has disposal of the ball for the throw-in. No longer able to correct error.
1. Rebound of missed FT. (Live ball and clock starts)
2. Made basket (dead ball)
3. Team B has disposal for Throw-in (live ball)
Correctable error has to be done before #3.