About 2 minutes after the tourney time ended, the score refreshed. I was alone on 577, and the next closest was something like 559. A couple minutes later the score refreshed again, and it showed the 3 way tie. Only one person was above me, and one person was below me.
If ties are broken by awarding the person who got there later, I should have been 3rd. I don't understand. If anyone were to actually know how it worked it would be someone like you, but the "latest wins" logic didn't apply from what I saw.
Perhaps it checks when battles started? Say, Lady Demeter had started her last match before you had started your last, but she took all 4 rounds to finish with a slower opponent. Thus giving you time to start your match and finish it before she finished hers. Therefore, going by just starting times, she would have scored before you when really you did before her.