Tie Goes to the Runner?

Recently, an interesting conversation ensued on the SABR listserv regarding the phrase "a tie goes to the runner". Is this true?

In the official MLB rules the only phrase I can find (repeated in rule 6.05(j) and 7.08 is "the defense must tag him [the batter] or first base before he touches first base."

To me this indicates that a tie does indeed go to the runner since the word "before" is used. However, several on the SABR site argued that umpires are taught that there is no such thing as a tie and so there will always be a correct call. Interestingly, one of the contributors to the SABR list noted that the "1875 DeWitt's Umpires Guide was emphatic that ties go to the runner: 'If simultaneously, the base runner is not put out...'". He also noted that when he umpired he was instructed that a close play was an "out that you needed to sell". Is that still the training for MLB umpires? If so, it would appear that umpires are in fact allowing ties to go to the fielder in contradiction to the written rule.

While training umpires that there are no ties may be necessary, in reality there are limits to both visual and auditory human perception. In fact, another poster noted that if two events occur within 0.05 seconds it is impossible to reliably judge which occurred first. He also notes that auditory reaction times are typically quicker than visual ones. And so in practice there are ties and so it would be nice if the rule book addressed the issue specifically.