Here's a slightly more complicated variant: if the vote is close (for any reasonable definition of 'close') the outcome is decided by a random process, but one that does not always have a 50-50 outcome; instead it is biased, in proportion to the difference in votes, towards the side with the higher count. Only if the counts are equal are the probabilities equal, and if the difference in counts is close to the threshold for invoking this mechanism, it will almost certainly choose the side with the higher count.
For example, a series of coin flips, with each win being added to the toss-winner's vote count, until one has reached a value one greater than the higher vote count. This seems to be a natural extension of the sudden-death tie-breaker.
Because the counts are discrete, I do not think there is any sorites paradox here, and so long as more than a few votes were cast, there should be a reasonable definition of 'close'.
This would never work in practice, as the side with the larger count will always regard it as unfair, but could perfectly rational participants find this acceptable?