They do, in fact, have a house! Problem is, it’s not named. If you look on Wikipedia, they get the title 皇室 (こうしつ, koushitsu), but the problem is, that just means “Imperial family” (and more literally it translates to “emperor’s room”).
As far as I know, this has always been the case, and hence the current Emperor of Japan, Emperor Naruhito, really is just called Naruhito, and nothing else. (Though if you were to call him just Naruhito to his face, expect to be in a very awkward situation).
Pictured: Emperor Naruhito (徳仁)
As for how this would translate to CM, uh…
Call the royal house Windsor皇室 from now on???
Er, really, we would have to know the actual identity of some the members of the TWE royal family to figure something like that out.