This is an interesting rule, so I'd like to know if I'm reading it right. It sounds like in the scenario above, if the Umpire does not judge that the player could have made it to second safely had there been no obstruction, then the player can be put out, but only if another runner (say the one running to first) is put out between the obstruction and the tag on the obstructed runner.
I think you're reading that almost right, except "make a play" on a runner does not mean getting the runner out, so you don't have to actually put out another runner. Just make a play on her (a throw to a base, for example).
First: It says is that once the obstructed runner reaches the base they would have obtained had there been no obstruction. Example: Batter hits a single to the outfield, rounds first, collides with the 1B who isn't paying attention. The umpire calls obstruction, but in his judgment, the runner was not actually going to get to second (hard hit ball, OF got it back in to second base quickly). In this example, the base they "would have obtained" is first base even though the obstruction happened between first and second.
Second: "there is a subsequent play on a different runner." Let's go back to the example. Ball hit to the OF. Runner rounds first, collides with 1B, umpire calls obstruction. A runner who had been on second tries to score. So after the ball comes back into second, there's a play at the plate. In the meantime, the runner on first jogs to second. Catcher throws down, and she's tagged out before reaching second base. She IS out, even though there was obstruction between first and second, because A) the runner had already safely reached the base the umpire deemed she would have reached had their not been obstruction and B) there was a play made on a subsequent runner.
Now, the reason you need this exception is that, had there been no play on the runner trying to score in my example, the runner is protected from being put out between first and second, and if she IS tagged out, she is awarded the base that the umpire judges she would have been awarded had their been no obstruction. In the case of a sharp single to the outfield with no errors getting it back in, that base is likely to be first base. It still reads to me that if the umpire believes the runner would have reached second had there been no obstruction, she'd STILL be awarded second even if there was a play on another runner. There's nothing in the rule that says a player is automatically awarded the next base though, and I think that confuses people at times.
I've read my explanation a whole bunch of times and I think it's properly interpreting the rule, but I'm not sure I explained it well. But it's possible I have no idea what I'm talking about.
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