Here's one possibility. The lantern may attach between the cover (the octagonal part at the top) and the fixture (the glass bits etc.). See this manual for a very similar if not identical fixture.
If that's the case, then either like that manual suggests there will be a couple of "decorative" nuts hidden that attach the two, or it's possible the fixture screws into the cover itself (less likely with a squareish shape, but who knows what could be under that). If the latter is the case, we're not talking a bell jar type screw, but one where it screws maybe a half turn from an "open" position to a "closed" position.
Decorative nuts can be hard to spot, since they're intended to be hidden, but there's likely 2 or possibly 3 - so see if there's some different thing either on opposite sides or in a triangle that might be hiding nuts. I see something that looks like it could be a nut on the left side of the picture.