Playing with wrapping #AllSkyCamera footage onto a curved surface in Blender to recreate what it actually films, just a quick try, but this has potential.
Do you need a dome, or just a room of any convex shape with high enough ceiling? Parallax shift rates (i.e. angle change per head position linear change) are inversely proportional to distance, so the shifts should be continuous but not necessarily smooth for a non-dome ceiling. I'm not sure what's the threshold for noticeability.
@robryk Interesting idea - could be done with two projectors and polarising filters/glasses as per 3D cinemas.
Yeah, ideally it needs to be on a nice smooth sphere, any angles are going to throw the layout off.