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



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:


  
  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.