When does Tiana, Ship's Caretaker check card type?
The ability will still trigger; what matters is what the object was when it was on the battlefield. 


  603.10. Normally, objects that exist immediately after an event are checked to see if the event matched any trigger conditions, and continuous effects that exist at that time are used to determine what the trigger conditions are and what the objects involved in the event look like. However, some triggered abilities are exceptions to this rule; the game “looks back in time” to determine if those abilities trigger, using the existence of those abilities and the appearance of objects immediately prior to the event. The list of exceptions is as follows:
  
  
    603.10a Some zone-change triggers look back in time. These are leaves-the-battlefield abilities, abilities that trigger when a card leaves a graveyard, and abilities that trigger when an object that all players can see is put into a hand or library.
  


Tiana’s ability is a leaves the battlefield ability; even though it talks about entering the graveyard:


  603.6c Leaves-the-battlefield abilities trigger when a permanent moves from the battlefield to another zone, or when a phased-in permanent leaves the game because its owner leaves the game. These are written as, but aren’t limited to, “When [this object] leaves the battlefield, . . .” or “Whenever [something] is put into a graveyard from the battlefield, . . . .” (See also rule 603.10.) [...]