|Globe program displaying output from Plate Tectonics simulator|
It turns out the distortion results from the vertex sharing required for auto-generating normals that make the polyhedron look like a globe. The fix turned out to be to deliberately introduce minute differences into the y coordinate of the vertex at the north pole, so each triangle at the top had a minutely different vertex position at the top but with much different texture coordinates. The details can be seen in comments in the source code.
The source code is available under MIT license at https://github.com/AnAvidDeveloper/Globe. I will look into making a binary available in the future.