Let us presume that existence and interaction can be separate. Let us also assume for a moment that there is a definition of 'existence' without interaction. Then a reality with a thing that exists but doesn't interact is equivalent to a reality where this thing doesn't exist. If a thing's existence and non-existence yield equivalent realities, we can always assume that it does not exist. Therefore, we can set 'it interacts' and 'it exists' to be equivalent for all practical purposes.
That's my line of thought, which of course still doesn't exclude that things can 'exist' but not 'interact'. But if you would press me, I'd ask for a definition of the word 'existence' without using any form of interaction. I wouldn't know a proper answer to that.