I wish I had been around on the game thread to clarify some things after the Reyes/Lawrie situation had occurred but I had to leave. I read the game thread when I got home later on last night and the posts today. I feel I need to clarify the situation and correct some misconceptions. This is something I do know about, I'll try to make it as easy to understand as I can.
First and foremost: The ONLY time a player can be tagged out when standing on a base is when two players are touching the base at the SAME time.
Clarifying a misconception: with the rule about "two runners may not occupy a base", that is when the play is not alive(a dead ball situation).
When during a live ball situation two players are touching the same base only the lead runner(Reyes) is deemed to occupy the base and the trail runner(Lawrie) may be tagged out. I'll use Reyes and Lawrie from here for ease of explaination.
Important to note: in this situation, at their own peril Reyes may try to advance to the next base and Lawrie may try to go back to the previous base. Also important, they are are not stuck there.
Clarifying a misconception: moving back to a base or away from a base has NO effect to the right to the base. As well, being 2 feet away from a base is the same as being 85 feet away from a base. Reyes being 2 feet off the base is the same as him running 85 feet away about to touch home. Reyes ALWAYS has the right to go back to the base(this is not a force situation)
Nobody had been touched yet, so when Reyes came off the base he is to be considered going to the next base(home). At that point Lawrie is the only player at third base and now can be deemed to occupy the base and CANNOT be tagged out.
The third baseman now tags Lawrie who should NOT be called out, he then tags Reyes who SHOULD be called out, as he is off the base. Lawrie never left third base so no matter how many times the third baseman tried to tag Lawrie he should have always been safe.
This is a play with lots of commotion so players and umpires can sometimes get confused with everything going on, the only way the umpire can really make this mistake is that he thought Reyes was touching third when Lawrie was tagged meaning Lawrie would have been out and when Reyes was off the bases he is tagged out. The result being both players out.
Kraemer_17 was almost right with the correct way the defender(it can happen at 2nd base as well) is to be taught in how to react in this situation. When two players are standing on the same base the defender is to tag the trailing runner, then tag the lead runner and say directly to the lead runner "You're out!". You do it this way because the trail runner is already out, but sometimes the lead runner can get confused and when he hears someone say he's out he can step off the base and then the defender can tag him out as well. It occurs very rarely but it DOES happen, it's even worked for me a few times.
In conclusion, the umpire made a mistake and if he saw the replay he would admit to such(probably)
There were probably a couple other misconceptions left out, but these I did from memory. I hope this clarifies things, I'm not the most experienced writer. And nice bomb Lind!