What rules explain Xantcha's behavior when its controller leaves the game?

Question

Xantcha, Sleeper Agent has the ability:

As Xantcha, Sleeper Agent enters the battlefield, an opponent of your choice gains control of it.

The release notes for Commander 2018 describe what happens when a player who controls a permanent owned by another player leaves the game as:

If that player controlled any permanents owned by another player, the effects that gave control of them to the player who left end. If this doesn't give control of them to a different player (perhaps because they entered the battlefield under the control of the player who left), they're exiled.

In the same document, the release notes for Xantcha state:

If the player under whose control Xantcha entered the battlefield leaves the game, the effect changing control of Xantcha ends.

Answers to a previous question lead me to believe that Xantcha's replacement effect causes it to enter the battlefield under an opponent's control and not under my control at all.

Because Xantcha appears to be entering the battlefield under an opponent's control, I thought Xantcha would be exiled when its controller leaves the game.

However, I know this is not what happens because the official Magic: The Gathering Rules Manager has answered a question about it:

What happens to Xantcha when the opponent controlling her dies? Does she get exiled? Returned to my battlefield? Do I get to give her to another opponent? Is all of this in the release notes?

It's all in the release notes, but I'll tell you! You end up with Xantcha. Will you keep her as your poisonous friend, or kill her so you can pop a new Xantcha out of the vat? #WotCStaff

So I know what is supposed to happen - Xantcha's owner is supposed to gain control of it when its current controller leaves the game. What I want to know is the full explanation within the rules of what causes this result to happen.

Answer 1

When Throne of Eldraine was released in October 2019, Xantcha, Sleeper Agent received functional errata that makes it now work the same way a card obtained with Bribery works; and Xantcha will now be exiled by 800.4a when her controller leaves the game.

Leaving the original answer below; as the question only existed due to the old wording on Xantcha and the old rules associated with that wording.

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Xantcha, Sleeper Agent has an ability that generates a continuous effect. This continuous effect then ends due to the first part of rule 800.4a:

When a player leaves the game, all objects (see rule 109) owned by that player leave the game and any effects which give that player control of any objects or players end.

We can see that it is a continuous effect:

611. Continuous Effects

611.1. A continuous effect modifies characteristics of objects, modifies control of objects, or affects players or the rules of the game, for a fixed or indefinite period.

611.2. A continuous effect may be generated by the resolution of a spell or ability.

611.2a A continuous effect generated by the resolution of a spell or ability lasts as long as stated by the spell or ability creating it (such as “until end of turn”). If no duration is stated, it lasts until the end of the game.

This is confirmed by a Gatherer ruling on Xantcha:

If the player under whose control Xantcha entered the battlefield leaves the game, the effect changing control of Xantcha ends. Because Xantcha’s replacement effect creates a continuous effect causing another player to control it, Xantcha isn’t exiled.

This is different from the wording on cards such as Bribery, which says "put that card onto the battlefield under your control". In that case, there is no continuous effect, the card simply entered the battlefield under your control. Because of this, the first part of 800.4a would not see a control-changing effect to end, and instead the third part of 800.4a would exile the creature.

Note that the wording that causes the controller to be an opponent is the same on Xantcha as it is on cards like Donate, "[...]player gains control". The only difference is that it is a replacement effect that happens as the creature enters the battlefield, instead of something that happens to a creature already on the battlefield. Bribery on the other hand has no such replacement effect or continuous effect, it simply lets you put something on the battlefield under your control.

It is still correct that you never controlled Xantcha in the first place, but that is not relevant. The rules don't cause Xantcha to "revert" to its original controller, they simply cause a control-changing effect to end. Theoretically, Xantcha could end up going to a different opponent instead, if there were multiple controller-changing effects in play.

Answer 2

Xantcha will be exiled when the opponent that has control of her leaves the game but the game continues.

The owner of Xantcha never had control of Xantcha, so there is no controller to revert to if the opponent controlling her leaves the game. Thus, Xantcha will be exiled. I have to suspect that the Rules manager tweet was mistaken, if he indeed meant that the owner will gain control of Xantcha. The bolded example of rule 800.4a is the closest equivalent to the Xantcha situation:

800.4a When a player leaves the game, all objects (see rule 109) owned by that player leave the game and any effects which give that player control of any objects or players end. Then, if that player controlled any objects on the stack not represented by cards, those objects cease to exist. Then, if there are any objects still controlled by that player, those objects are exiled. This is not a state-based action. It happens as soon as the player leaves the game. If the player who left the game had priority at the time they left, priority passes to the next player in turn order who’s still in the game.

Example: Alex casts Mind Control, an Aura that reads, “You control enchanted creature,” on Bianca’s Assault Griffin. If Alex leaves the game, so does Mind Control, and Assault Griffin reverts to Bianca’s control. If, instead, Bianca leaves the game, so does Assault Griffin, and Mind Control is put into Alex’s graveyard.

Example: Alex casts Act of Treason, which reads, in part, “Gain control of target creature until end of turn,” targeting Bianca’s Runeclaw Bears. If Alex leaves the game, Act of Treason’s change-of-control effect ends and Runeclaw Bears reverts to Bianca’s control.

Example: Alex casts Bribery, which reads, “Search target opponent’s library for a creature card and put that card onto the battlefield under your control. Then that player shuffles their library,” targeting Bianca. Alex puts Serra Angel onto the battlefield from Bianca’s library. If Bianca leaves the game, Serra Angel also leaves the game. If, instead, Alex leaves the game, Serra Angel is exiled.

Example: Alex controls Genesis Chamber, which reads, “Whenever a nontoken creature enters the battlefield, if Genesis Chamber is untapped, that creature’s controller creates a 1/1 colorless Myr artifact creature token.” If Alex leaves the game, all such Myr tokens that entered the battlefield under Alex’s control leave the game, and all such Myr tokens that entered the battlefield under any other player’s control remain in the game.