I have a service dog for my dissociation, I have a short explanation that I give anyone who asks. "You know when you think of something you want from another part of the house and when you get there, you can't remember what it was you needed? That's dissociation, a normal thing. But I can do that for hours or even days." When people ask why I do that, I just say that I don't really know. I don't mention alters
That's what I have told my congregation (I am a pastor). The dog is the only accommodation I have asked for.
It works for me because it is truthful while protecting my privacy.