The Trickery Domain cleric gets the Channel Divinity: Invoke Duplicity feature at 2nd level, which creates “a perfect illusion of yourself” (PHB p.63).
However, CD:ID’s description doesn’t really tell us anything about what this illusory double actually does while it lasts (does it mimic its originator’s actions, or does it act independently?), nor do the rules tell us whether this duplicate can be revealed to be an illusion — using an Intelligence (Investigation) test, perhaps, or via a physical attack? — and if it can be, what happens if someone “sees through it”.
Have I missed something, or are the answers completely up to the DM? (I think the latter is perfectly fine, btw, just don’t want to houserule something that’s covered officially.)
The rules only cover combat (well..casting) actions for this ability and do not really speak to how to use this outside of a combat situation.
Here’s what we do know about it:
You can use it as a point of origin for casting (Though you still have to be able to see the effect point as it uses your senses, not the duplicate’s)
You get advantage if it’s in melee with you.
The problem here is that it’s a bit of an odd thing for it to not speak to whether or not it can be A. attacked and B. disbelieved.
As far as A goes, it should be able to be attacked, but obviously it’s an illusion, so it’s not going to take damage. The concern I’d have is a situation where you and the illusion are both valid targets, how does an opponent decide (as a DM, I’d flip a coin unless the monster has a good reason to know otherwise (like they saw you cast the spell just a moment ago).
The question of whether or not it can be disbelieved is far more complicated. The key word here seems to be “perfect,” if it’s a perfect illusion then it would seem that it would quite hard for it to be disbelieved. Perhaps if a monster had reason to disbelieve it (for instance, was interacting with it in some way), they could make a wisdom saving throw against your spell save DC possibly at disadvantage (it depends on how strongly you want to take the word “perfect”).
Ultimately though, all we have to go on are those two bullet points above. So this is squarely in the land of DM discretion.