Visitors are down by one run in the top of the ninth with one out and runners on second and third bases. The batter hits a scorching line-drive down the right-field line. The runner on third comes home to score, and the runner on second is rounding 3rd when the right-fielder makes a spectacular diving catch, and then throws it to the shortstop covering second for a game-ending double-play.
The crowd goes wild, and the shortstop throws the ball into the stands in celebration.
The visitors' coach says
Wait a minute... our runner scored way before you got the third out at second. It doesn't even matter that he didn't tag up on third because once you've gotten the third out, the inning is over, and it's too late to appeal. Had you wanted that run not to score, you should've thrown it to third instead.
Besides that, you've thrown the ball out of play, so even if you were allowed a fourth out, you couldn't retrieve it and appeal because the play is officially over once the ball leaves the playing field.
He looks around to find the scoring-runner to have him go back to third and score correctly just to be sure, but that player has already hidden somewhere private to cope with the emotions.
The home team's coach replies
You can't score an illegal run like that. It doesn't matter which bag we throw it to, he still has to tag up. And even if you ever do get them to tag up and score, that run still wouldn't count because he left the baseline a long time ago.
He yells at his players to stop celebrating and tells them to go get the ball out of the stands so that they can appeal to third.
What is the right call? Does the run count and the game moves on to the bottom of the ninth? Or does the home team win? Or do you wait to see if the runner tags up and scores before the ball is found and appealed at third? Is a fourth out allowed in situations like this? And if so, how can you be sure that they found the right ball?