Eh, it should use pole vaulting/high jump tiebreakers :)

That is, on the turn any empire unlocks Era IV, among all nations that tie at 100% approval, it should search backward turn-by-turn through the game's history of nation stats per turn, and eliminate contenders when their approval drops below 100%, until only one is left.

That is, it should go to the empire that achieved 100% approval at the earliest time, and has remained there ever since.  That's usually Cultists, since they'll typically hit 100% in Era II when they build Central Market, and there shall they remain forever more.

If there are still 2+ nations that tie, i.e. they all reached 100% in the same previous turn and have remained there ever since (possible with 2+ Cultist players, haha), then it could go strictly by the player sequence list (that you see when you hover over the End Turn button).  Maybe it does.  It seems to do that already for legendary buildings, and the human player is always #1 in the player sequence, so if you're 1 turn away from completion, you know you've won it because you'll win any construction "tie" at the start of next turn.