From the little ball board...
 

So they are having an interesting discussion there about 4th outs and I wanted the NFHS take. (In ASA this can't happen on an allowable appeal.)

R1 on 3rd, 2 outs.

BR hits a near double but is tagged out at second. On the way to second he missed first base.

Question: Is he allowed to attempt to retouch and correct his base running mistake even though he's out? I'd think not but can't come up with a rule to back me up. So what's the rule support?
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PRILOSEC DEATH

Seems to me that common sense would dictate that only a runner can legally run the bases and if a player is put out, they are no longer a runner.

Agree with Irish, once he is tagged out for the 3rd out the ball is dead and he can't return to touch a base missed. Once he passed 1B he is assumed to have touched it unless properly appealed.

However, if the Defense appeals that the BR missed 1B then I believe you have to honor the appeal, as long as the dead ball appeal is made properly. At that point, it is a force out and no run scores?

In baseball, the ball is not dead on the third out since an advantageous fourth out appeal can happen and an appeal can only be made with a live ball (exception: NFHS).

No run scores in NFHS. In ASA you don't honor the appeal.
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LIVE SEX

However, the original question did not concern the appeal, but the ability for a retired runner to return and touch a missed base.

Sure. I don't see how this is really possible. The runner is out and no longer a runner, but a retired runner. I wouldn't allow the runner to return, but would uphold an appeal. Just cause the defense put him out doesn't mean they should be put at a disadvantage. If the runner was safe at second, the defense could appeal and the run wouldn't score, after all.

why is the ball dead? Is it just because the inning is over?
What if the defense plays on another runner, gets that out, and then the original presumed third out is overturned?

Ball is dead because the inning is over. They can't make a play on another runner. They can make an appeal for the 4th out but only a dead ball appeal. You can't tag out another runner for a 4th out/insurance out.

Rule 10 does allow for an umpire to rectify a situation were a runner was placed in jeopardy. Once that umpire called the 3rd out he has placed all other runners, and defense for that matter, in jeopardy so you won't honor that "4th out" play on another runner, just like you won't allow a runner to score after the 3rd out.