For the US Open Championship, there is a tie-breaker provision in the rules. In the event of a tie during the tie-breaker round the provisions state:
"In the event of a tie during the Championship tie-breaker round, the
better scoring entrant between or among the tied entrants will be the
entrant with the lowest combined score for the final six (6) holes of
the tie-breaker Game. If any ties remain, the better scoring entrant
between or among the remaining tied entrants will be the entrant with
the lowest combined score for the final three (3) holes for such Game.
If any ties remain, the better scoring entrant between or among the
remaining tied entrants will be the entrant with the lowest score for
the 9th hole in such Game, and any remaining ties will be determined by
identifying the lowest scoring entrant for each individual tie-breaker
game hole, in sequence from 8th through to 1st, as is necessary to
determine a winner and, if any ties remain at that point, the
earlier-submitted entry between or among the tied entries will be
deemed the better scoring entry."
I'm pretty sure that the majority of that text was copy and pasted from your 9 hole tournament rules page, and is slightly off for an 18 hole tournament format. Should it not read as follows?
"In the event of a tie during the Championship tie-breaker round, the
better scoring entrant between or among the tied entrants will be the
entrant with the lowest combined score for the final six (6) holes of
the tie-breaker Game. If any ties remain, the better scoring entrant
between or among the remaining tied entrants will be the entrant with
the lowest combined score for the final three (3) holes for such Game. If any ties remain, the better scoring entrant between or among the
remaining tied entrants will be the entrant with the lowest score for
the 18th hole in such Game, and any remaining ties will be determined by
identifying the lowest scoring entrant for each individual tie-breaker
game hole, in sequence from 17th through to 1st, as is necessary to
determine a winner and, if any ties remain at that point, the
earlier-submitted entry between or among the tied entries will be
deemed the better scoring entry"