TL:DR press Set From Faces in the Shading tab of the Tool Shelf.
What's happened is your normals have got bent out of shape in a weird way that the usual Ctrl + N make normals consistent doesn't see as a problem.
They're all facing the right way vis-a-vis inside/outside, but they're all over the place in terms of direction. So you just tell Blender it needs to force all normals to be perpendicular to their corresponding face.
I still have no idea how this problem occurs in the first place, though. It foxed me for a good few days.
