A topic from AEG's L5R CCG Rules forum, copied for Oracle of the Void from a copy of the forum saved in October 2015, so pages it didn't save are missing. Players' names are left out; the rules officials are named. Rulings are as they stood on these dates; later rulings, the rulebooks and the Rules Archive may differ.
A player, 2 June 2011
Kobi’s ability reads:
Reaction: After another player's Battle action resolves which destroyed one of your Lion Clan Personalities at the current battlefield: During the next End Phase, if the destroyed Personality is dead, bring him into play, ignoring costs and Honor Requirements.
Must the use of the reaction be declared immediately after the destruction of said Personality, or may a player wait until the End Phase and choose (from those who have been destroyed during the previous attack phase) which one returns to play?
Zen Faulkes (rules official), 2 June 2011
Akodo Kobi (Experienced) Must the use of the reaction be declared immediately after the destruction of said Personality(?)
Yes. Just like every Reaction, the trigger "after" = "immediately after."