fly ball interference
 

I think we answered this last year or before; but I can't find it.
USA Rules only, NFHS is very different; probably others.

If a runner interferes with a fielder attempting a fair or foul fly ball which is catchable with ordinary effort; the runner is out and the batter is also out.
8.7.J.3 (1) effect F in the wrong place

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Even if there were two outs before the play, the batter has completed a time at bat, so does not bat in the next inning.

Is that correct, or does that batter appear again?
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Opinions are no help, already have both sides from very learned sources; need a case or play/clarification documented.

There is no case play from the 2017 case book provided at the USA Softball Biennial clinic. I do not believe there has been anything issued in the clarifications either.

Let me offer this: you mentioned that in NFHS the rule is different; the effect is different in USA Softball if the runner interferes with a fielder on a ground ball. In both cases, interference is called and the batter is placed on first. Both cases result in the batter completing their time at bat.

Therefore, why would this instance be any different? In all cases of interference in USA (and fair ball interference in NFHS), the batter has completed their time at bat.

In USA softball, if a runner interferes with a fielder on a pop fly ball, fair or foul catchable with normal effort both the runner and the batter are out. It is specified in the rule as well as the rule supplements. Has been that way for many years, and yes nfhs is different.

But USA does not allow a fourth out unless it is an appeal on a runner who has scored. There is no doubt the player became a batted runner and completed their turn at bat.

Now the conundrum. It has been suggested elsewhere that the player in question be the lead-off the following inning. IMO, that is not possible by rule or common sense. To allow that player to bat again, you must ignore the previous action. If you ignore that pop fly in the previous inning, you have no ball put into play and in turn, cannot have INT and if you did not have INT, the previous inning never came to a proper end. :)

Sorry, missed the second half of your question. Didn't read down far enough.

Does not allow a fourth out? Or does not allow an appeal for a fourth out?

Along the lines of Irish's rationale, it was the batter's putting the ball into play that made the 3rd out possible (unlike a runner leaving early on a 2-out called 3rd strike).

Neither

Allows 4th out appeals, but ONLY against a runner who has scored a run.

Yes, I understood that. But, my question was about the rule that "does not allow a fourth out unless it is an appeal on a runner who has scored."

Sometimes, when you have rules-lawyers reading rule books and trying to apply a loophole or gotcha to the rule as written instead of the obvious spirit and intent, you have to turn that against them. If the rule specifies that appeals for fourth outs are not allowed (with the exception), then you can make the argument that this is not an appeal. The batter is out by rule and you simply have a fourth out that inning.

IRISHMAFIA says this is not the case and the rule apparently forbids all fourth outs (with the exception).

You do not have a fourth out, by the rule that Mike quoted. But it doesn't matter.

In USA Softball, it doesn't matter how many outs, the interference on a batted ball ends the at bat. Period. End of story. Don't complicate this rule.

Look at the other rule codes for comparison:
Over fair territory:
NCAA: rule is the same as USA - runner and batter are out
NFHS: runner is out, batter is put on first base
Bottom line: in all cases, the batter's time at bat has finished when the interference happens on a ball over fair territory.

Over foul territory:
NCAA and NFHS: runner is out and a foul ball (strike) is called on the batter. Batter remains at bat. If the third out of inning, current batter would lead off.
USA: batter and runner are out. This is different than the other two codes, however, the same logic applies as in a fair ball - the batter has completed their time at bat.

*sigh* I was simply trying to provide an alternative answer to somebody who wants to complicate the rule. You can say this all you want, but some people simply will complicate it without a case play or some other clarification. The OP is asking for an explanation to give to those people.

My thinking is that the batter was put out by the interference, by rule, and so has completed the at bat.

Otherwise, saying batting again ignores the out, but both outs occur simultaneously; so ignoring he batter’s out would ignore the runner’s out.

IOW, either when the batter is out or when the batter is awarded first base, the at bat is over.

What part do you think isn't clear?