There's no difference between an actual, physical, material, governing office and a spiritual office. It is all one and the same. The distinction was invented by the heretic des Laureirs. No real Catholic, much less one who called himself a Traditional Catholic, would give this novelty a second thought.
While not quite the same distinction, Bellarmine made a distinction between the matter and form of the papacy, i.e that one can have the matter of the papacy(sitting on the throne of St. Peter and being recognised by the faithful as the pope) without having the form(actually being pope). Someone with the matter but without the form(that is, they lost the form due to heresy or some other means but they have not been deposed yet) would de facto occupy the material, governing office but not the spiritual office. At least in theory.