So I want to discuss a very specific topic as an example of what it means to do "realist" ecclesiology.
Now, it may seem pretty obvious that of course the pope controls what it means to be Roman Catholic, but I came across a rather interesting incorporated church in the Californian business registry named HOLY ROMAN CATHOLIC CHURCH - PATRIARCHATE OF AMERICA. Apparently someone incorporated an entity with that name. In realist ecclesiology, we're not concerned with abstract or in theory ecclesiology about the pope's claims of control, we're concerned with on the ground material and physical causes and effects, and enforceability to such claims. Can the Pope sue to disincorporate this entity and stop them from using or claiming the name?
Consider another more interesting angle: could someone simply trademark the name "The Roman Catholic Archdiocese of Hollywood"? Then that person, or corporation, would in effect be able to use and own the name and there's nothing the Pope in Rome, or even the Archbishop of Los Angeles, can do about it. They don't have the material means to control who claims and uses the name.
So in effect, without the necessary political or material power to make good on his claims of universal jurisdiction, he can't really control what's literally in the name of "Roman Catholic" even, it is ultimately subject to the will of the civil authorities.
I'm not the most familiar with trademark and patent law, but if I recall correctly, one of the main standards for judging trademark infringement is whether the phrase or image or whatever could cause confusion that the person using it is the holder of the actual trademark.
So as an example, I probably can't make a video game also called Starcraft since this would cause confusion, but I could call a telescope manufacturing company that because it's a completely different realm of business so unlikely to result in confusion by general consumer.
Let's assume for the sake of argument that the Pope has a trademark in the US on the phrase "Roman Catholic". If he does, then I think an unaffiliated entity using the name "The Roman Catholic Archdiocese of Hollywood" would probably be violating the trademark (if true, I'm not sure whether this means they would be unable to trademark this name or whether it would simply be a worthless trademark because they would be injoined by a court). I'm not an expert though, but I think difficult to get around the trademark here (assuming that the Catholic Church really has trademarked some variation on its name and did so in a timely manner).