I found someone with similar problem, but their answer doesn't work for me. Other than that, the net is full of advice how to flip normals programmatically to create interiors on the fly, but other than "Your normals are wrong", no help.
All my normals point outwards. I exported the object to .obj, imported it into Unity, and got:
Did I apply the texture to the reverse side of the faces, or something like that? How do I fix it?