Well, this question seems to have been lost in the insurance discussion below, so I thought I would raise it up a level as I really am interested in what people would do in this situation.

Consider this: R1 on 1st, 0 or 1 outs, BR hits the dying quail to F4, BU calls Catch, Out; PU calls No Catch. R1 and F4 respond to BU's call, R1 holds at 1st and F4 throws the ball back to the pitcher. Savvy 1st base coach hears the PU call No Catch and calls the BR back to 1st and has her stand on the bag with R1, then looks expectantly at the PU. So you as the PU, realizing that you have a 10-6-C situation on your hands, reluctantly call time and meet with the BU. The BU rather sheepishly admits that he is not 100% sure that there was a good catch while you are 100% sure that you saw the ball skip into the glove. So armed with this information, you leave the huddle and your ruling is:<ol><li> the catch was good and the batter is out,<li>the catch was not made and the BR gets 1st with R1 going to 2nd, <li>the catch was not made, the BR gets 1st, but R1 is out since F4 would have had an easy plsy, or<li>None of the above - so what do you do instead?</ol>

SamC

Well, since ASA did give us 10-6-c, might as well use it. Runners on 1st and 2nd, no out on the play. Now, stand back and brace yourself for a defensive demonstration of "wailing and gnashing of teeth." When this is over, maybe another discussion of mechanics is due with the BU.
R1 was placed in jeopardy when the out call was heard from the BU and they went back to 1st. A definite No-No. Of course, you took away a possible chance for an out for the defense. They deserve the right to express their displeasure, for a few seconds.

My answer is 2.

If the play was called correctly, it would have been no catch and R1 would have been forced to 2nd. We may (required in NFHS, not in ASA) to rectify problems caused by our mistakes. However, that applies to placing the offense or defense in jeopardy.
The umpires did not place R1 in jeopardy unless F4 was in a position that might have made an out impossible. Ignoring that for now, they did place the defense in jeopardy as indicated by F4 hearing the "out" call and returning the ball to F1. Therefore, R1 should be out and the BR get 1st on the non-catch.
If F4 was in a position that might have made an out impossible, then the umpires must judge who was placed in jeopardy by the wrong call. IOW, would F4 have tagged or forced R1 or would R1 have reached 2nd safely.

say to my partner, look, if you hadn't jumped on my call we would have BR safe at first, and most likely R1 out on a tag or force. But, I'll never be able to sell awarding an out that wasn't earned. So, even though your call, which you shouldn't have made anyway, was wrong, we're letting the OUT call stand. The defense gets the out they should have gotten, and the offense still has one runner on 1st.

OK... not entirely serious; but maybe I am!

In a FUBAR'd play like this one, I've got BR on 1B and am putting R1 on 2B. Since I'm making the ruling, I'll put up with a little bit of stuff from the defensive coach and will not allow the coach to speak to my "partner". My partner is buying all of the after game refreshments and is going to leave that get together with a very large new opening in his backside.

One way on another, the defense has got to earn the out, and they didn't - no catch & F4 gave the ball to F1 instead of touching 2B. I can't get R1 for abandonment - never left live ball territory.

I'm not ASA. What's 10-6-c?

10:6:c

The plate umpire may rectify any situation in which the reversal of an umpire's decision or the delayed call by an umpire places a batter runner, a runner, or the defensive team in jeopardy. This correction is not possible after one pitch, legal or illegal, has been thrown, or after the pitcher and all infielders have vacated their normal fielding positions and left fair territory on their way to the dugout or bench area, or on the last play of the game when the umpires have left the field of play.

By the way TwoBits, as a teacher, I really appreciate your signature!

In NFHS, it's 10-3-m (see also 10-3-g), which does not include the word "may".