Have you found a satisfying end condition? The 13th eagle card never seemed like a good note to end on for me.
Still, it's a great game.
We modified that a bit, the game requires a specific number of game rounds equal to the number of players "+1" (more or less) - the game can only end in the round following the required number of game rounds.
Once the number of required game rounds have been completed, the game ends instantly if:
a. 1 player is eliminated
b. The Nth Eagle card is played, where N = #Players
c. the end of the N+1 Game Round is reached