Thanks, Rickei, but after having tested the same fbx file in a couple of other places, also exporting as an OBJ file from Carrara, running that through an FBX converter and ending up with the same result, I have to conclude that the problem lies within Carrara. So either I am doing something wrong here, or there is somethin wrong with Carrara.
Here is what I got so far in EIAS, pretty much straight out of the box. (The eyes are screwed up, and the pose is missing.) So close, and yet....so far