Flipping a coin is one solution; abolishing caucuses is better.
"Recount!"
You recount and it doesn't take long because there aren't a lot of ballots, but you take your time to be careful and you get the same number again, twice.
"Run off!"
But a run off is used to produce a majority for one candidate when the vote is split too many ways. In this case, it would be the same two candidates running again. That's not a run-off, it's a do-over.
Which is, you know.
Essentially random?
Holding the whole election over again until you get a clear result is expensive and inconvenient and not actually more democratic than the coin toss, under that situation.
A random result always catches folks off guards, but it's not like people make it up.
High card draw is popular.
Or perhaps, more appropriately, that we make a Constitution check.
Or my twenty five cents.
If you don't want to roll dice or draw cards or flip a coin, though, you'll probably just be left grasping at straws.