Your answer is marvelous and I will remember it for use when talking baseball trivia. It is also highly unlikely. If it were not unlikely, then quite a few baseball fans would know the answer to the question because we would see it regularly. That's what makes your question such a great one: it combines a rare fact situation with a rule known by few. It always amazes me that so many situations are covered by the rules even though you'll never, or hardly ever, see them. Just as it amazes me that somebody figured out that 90 feet was the correct distance between bases, or 60 feet, six inches, from the rubber to the plate. Move bases in five feet and batting averages would go up 30 points. Move the mound in five feet and nobody, repeat -- nobody --, could have hit Bob Gibson (that, and we might have a list of several pitchers killed by line drives).
The fact that your question is such a great trivia question is due, in large part, to the fact that it is so unlikely.
Your original post only asked for a way that a team was required to record four outs or suffer the consequences. I described a situation that's covered by the rules.
Is my scenario unlikely? Yes. One of the reasons it's unlikely is that several things must occur in a precise order, and some of those things are very unlikely. It's like an unassisted triple play -- luck, combined with some unlikely events. Believe Micky Morandini (didn't look up the spelling; just know he was with the Phillies) recorded the last UTP from second. Know that Dr. Srangeglove was the last one before that, from third base, in 1968, with the Pirates. In both cases, the hit-and-run was on.
Both Morandini and Dick Stuart TAGGED a runner for the third out -- a runner who had left his base (first, for Morandini, and second, for Stuart) before the line drive was hit and caught. Both Morandini and Stuart COULD have thrown the ball back to the base behind the runner (back to first for Morandini, back to second for Stuart). The runner would have been out. They didn't make the throw because the runner was right there (and fell down trying to make it back to second, in the case of Stuart's UTP, if I remember correctly) and tagging was easy, with no chance of an errant throw or misplayed ball. "General klutziness," as you correctly called it, happens in baseball, like a runner falling down; it just happens infinitely less in MLB than it does at a family reunion softball game. Lonnie Smith (on a delayed steal) not being able to see the ball against the Homer Hankies in the Braves/Twins World Series (and inexplicably not looking to the third base coach for direction as to whether he should run to third). The Ed Armbruster play in an early 1970's World Series between the Reds and the Orioles, when the catcher held the ball cocked back in his throwing hand and tagged the runner "out" at home with an empty glove (and getting away with it). General klutziness happens -- MLB lore is littered with heavy-hitting outfielders for whom every fly ball was an adventure.
Almost any infielder will tag a runner (or run over and step on a base), if that play's quickly available, rather than throwing. It's certainly what they're taught from T-ball through high school and American Legion and up to The Show. Tagging (or touching the base) doesn't risk the errant throw, the misplayed catch (or Reggie Jackson throwing out his hip to block the throw, as he did in a World Series game, the jerk).
My scenario is unlikely, but it's not impossible and it's not absurd. If the second baseman has an immediate chance to tag the runner from first to second, and it's a bang-bang play with the runner from third reaching home, he'll likely tag the runner and try to beat the play at home. Then he'll throw the ball to first. If he doesn't tag the runner and instead throws the ball away (imagine Chuck Knoblach double-clutching at the end of his career, when he couldn't throw to first, or those Sports Center moments when somebody with a multi-million dollar contract, for no reason, just throws the ball away), he'll get his rear end chewed out by the manager. During the last thirty or so games Chuck Knoblach played second, he would have done ANYTHING to avoid throwing the ball to first.
What's great about baseball -- and different than football and basketball for some reason -- is that everything will happen that can, eventually (except, perhaps, for the Cubs winning the World Series). That's why there are such great baseball trivia questions with answers involving unusual fact situations and rules that are rarely called into play -- like yours -- or just unlikely facts -- like mine.
You only asked for a way, within the rules, for a team to be required to post the fourth out to avoid adverse consequences. I came up with one that just happens to be different from yours.
I think your question and answer were great, but there is more than one answer to the question. Your scenario is far more elegant than mine. Yours brings into play a wonderfully unknown rule that will break the brain of those guys who are convinced they know all the rules (I certainly don't).
But if we're playing within the rules (in answering a question about playing within the rules), my answer, however unlikely, can happen and would result, within the rules, in the fielding team having to post a fourth out to avoid adverse consequences.
Your scenario has never occurred to my knowledge. I'm not criming you at all -- I said right off it was a clever thought -- and perhaps some day your scenario will occur.
But if the 3rd out in an inning is ANY sort of forceout, it matters not whether the runner on 3rd crosses the plate before the force is made.
I might have one real way it could happen. Say there are one out, bases loaded. A fly ball is it to the outfield. They get the guy out at third in a pickel as he tried to tag up, but they rule the runner scored first. They then appeal that the guy on first left too soon and win the appeal, which I think would disallow the run, although I am not certain on that.