Over here I'm pretty sure that's the only rule. Having whoever came there first go sounds like it could get very confusing in even vaguely ambiguous cases.
In my observations in my state, the "who gets there first goes first" is very much not confusing. What seems to confuse people is that with ties, people forget if the rule is the person to the right goes first, or to the left. So in practice, this tends to be negotiated using hand signals.