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, 10 January 2013
If a card, such as Akodo Kobi, refers to "your personality," does that refer to a personality that you own, or only those that you control?
Say I've used the Black Pearl to bring Akodo Kano into play in my home out of my opponent's discard pile. Then, in battle, I use the Ring of Fire, targeting Kano, destroying him and another personality. Could my opponent use Kobi's reaction, "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." to bring Kano into play in his home?
Is a card "your" card if you own it, or if you control it? Does ownership supersede control when a card is referred to as "your" card?
Ben Higgins (rules official), 10 January 2013
When “your” and other possessive forms such as “the player’s” or “his” refer to a card, this means “controlled by that player" if the card is in play, or "owned by that player" if the card is not in play. EXCEPTION: A trigger occurring after a card enters or leaves play checks the state of the card at the point just before the card left or entered play. See Timing: Triggers, conditions, and leaving and entering play.
The use of "your" to refer to a card does not override restrictions on effects, abilities, and targeting due to the card being out of play. Example: The trait, "Your Hero Personalities have +1F for each honorably dead Hero Personality in your discard pile." This does not mean that Hero Personalities you own out of play gain the Force bonus, because the effect only applies to Personalities in play by the rules on effects in and out of play.
When “your” and similar words refer to an effect, action, or targeting, they refer to effects, actions and targeting from that player’s cards, or from abilities or traits directly granted to the player. See Glossary, From.
To answer your specific question about Kano and Kobi, No, your opponent cannot use his Kobi's Reaction on the Kano you "borrowed" with The Black Pearl due to the Exception listed above.