It isn't alignment. I'll tell you exactly what it is.
The same thing happens when you tile multiple textures across one group of triangles.
Quads (at least, the way Saturn does them - but I think it's a universal thing) can't have textures applied across a large area... it's one quad -> One texture piece. So, when Satourne tesselates the quads into triangles, it tiles the texture where the quads would be... and you can't can't bilinear filter between two textures. The lines are where the bilinear filtering just stops.