Alright, here's the answer that I have. If you don't like it, feel free to argue amongst yourselves, because I've heard all the arguments in the world. But wherever it is that I read this (which I've conveniently forgotten), says this would work.
You got 1 out, and runners on first and third. The batter hits a long fly ball to deep center field. The runner on third is going to wait to see if it's caught, and then tag up to score. So he's standing on third. The runner on first does the deal where he goes most of the way to second, maybe even TO second, to see if it gets caught. The ball is, in fact, caught (for the second out). The runner on third, however, leaves the bag a split-second early, and runs home. The runner on first (who is now standing near second base), runs back to first. The centerfielder knows that guy is slow and tries to gun him down trying to get back to first. But the relay to first is wild, and goes over the first baseman. The runner slides safely into first, and then decides to run to second on the bad throw. Meanwhile, the runner who was on third has crossed home plate and is heading to the dugout. The first baseman recovers the ball, and throws the runner out at second base, for the third out. Inning over, right?
The run supposedly counts because no appeal was made at third base. The fourth out comes if an appeal was made at third base after the third out was made, which would make the run no longer count. Because no one is out on a force-out, the run counts as long as he scored before the out was made, and if no appeal is made at third base.