Continuing the previous talks, we will show how to discretize surfaces with a Finsler metric that is directed. (That is, not necessarily equal to the reverse metric.) We propose to imagine the following discrete picture: at the small scale, the surface is made of triangles with directed sides. Going along an edge costs one unit in the forward way and zero in the reverse way. We will see how these "fine structures" generalize square-celled surfaces, that were used to discretize self-reverse metrics. We also discuss the duality between fine structures and Postnikov's plabic graphs, and how the Postnikov strands play the role of the walls. Finally, we show a connection between discretizations of the 2-torus and integral polygons.