So I tried stacking an EF 2X III and an EF 1.4X III with an EF 12 II extension tube between them. The front of the 1.4 was extremely tight going into the rear of the extension tube, I had to force it. When I disconnected, a plastic piece came out. Now they go together and work just fine and I'm wondering if this little piece of plastic with a too small hole in the middle is even necessary.
Looks like it might be there as a baffle, to keep stray light from passing through that is not productive for imaging (flare).

