In regards to J.R.R. Tolkien and confusion as to what an anarcho-monarchist is, from my understanding it is a system where a monarch exists with it being an inherited position, but is mostly a figurehead who is used as final arbitrator in most cases, and could be a leader of military operations, but for the day-to-day living of the people within said society, there is no entity that would be recognized as a central government, i.e. "The State" that's my understanding of what it means of course.
I like to eat. Instead of a monarch I propose we have a Chef be final arbiter in matters. We'll call it anarcho-chefism.