Can I use Eater of the Dead's ability to exile any of the original Eldrazi titans in response to their ability to shuffle their owner's graveyard?
Yes.
The shuffle trigger of the original Eldrazi titans goes on the stack and can be responded to like any other triggered ability. This means that you can target Emrakul or Ulamog or Kozilek with Eater of the Dead’s ability in response, and Eater’s ability will resolve first and exile the Eldrazi before the trigger resolves (if Eater is tapped).
Note that the shuffle trigger will still be on the stack and will still shuffle the rest of that player’s graveyard into their library when it resolves.
















