I'm sure this question has been asked a million times but the watching the Tigers win on a two-run, walk-off home run made me wonder: Suppose either the batter or runner had missed a base? How does the appeal play work when the game is supposedly over, the home team is celebrating and the umpires are leaving? What does putting the ball back in play even mean in that situation?
I am going to try to explore as many possibilities as I can think of. (And I'm sure others can chime in.)
Suppose either the batter missed first base or a "forced" runner missed the base to which he would be forced, AND an appeal would have been the third out. Then, no runs score because you will not score a run on a force for out #3 or if the batter does not legally attain first base. If less than 2 outs, then the runners who are called out on appeal won't score, but the others will. This reminds me that one may also take an "advantageous fourth out" on an appeal to accomplish the same.
If a runner or batter-runner failed to touch some OTHER base, then their out would be registered, but runs may score. That's because you no longer have a force play situation. It's a timing issue And then you have to figure out who was in front of whom! (That might matter.)
In any appeal case, I sense that the gist of your question is "when can an appeal NO LONGER be attempted?" When the defense has left the field (often ruled as crossed the foul lines), then the half-inning or game is over, and no other play/appeal can be made.
Now if the defense correctly stayed on the field, and they wanted to appeal a missed base, here's what you have to do: The defense must all be positioned legally (i.e., battery set and the rest of the defense in fair territory), they need a baseball in the pitcher's hand, and then the umpire must put the ball back into play. (Note that you might not need a batter to come out and take his position in the box--only need that to start the game!) The ump says "play!" and then the appeal is made and judged accordingly.
I have seen this very thing happen in Little League World Series games and NCAA baseball CWS.
Hope that helps.
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