Inching away from symmetry

This system I call "Extrinsic Vertices" is great for generating symmetry, but slightly more difficult to manage with asymmetry. This image began as a relatively complex tile patch, with seven tiles. I'm fairly certain the tile patch couldn't be extended to fill the plane, without some significant rearranging. The biomorphic look I'm going for is one with symmetry changing to asymmetry and back.