I live in and own an end-of-terrace mews house in central London. I have a party wall with one next-door neighbour but my rear wall is entirely on my property and there has been nothing but a tiny garden there until recently.
Is my rear wall a party wall or not? It is entirely on my own freehold, is part of the structure of my house and pre-exists the construction of this garden area. If it is not a party wall, I believe we move into boundary law and trespass issues? Can next door fix anything to it without my consent?
If the wall is entirely on your own land and is not enclosed on the opposite face then it is your wall. You are therefore correct to say that nothing should be fixed to it without your consent. I would start by asking the Adjoining Owner to abate their trespass i.e. remove anything that they have fixed to it but should they refuse you will have to take legal action.
It is unlikely that screw holes would cause any damage but if the trellis is used for climbing plants that attach themselves to the wall they will cause damage in the longer term.