Suppose we take a blank puzzle and color around each cut differently, making bands. This essentially turns the puzzle into an edge-matching puzzle. Also, it works as a way to sticker spherical puzzles, which have no edges to divide the puzzle into faces. (The idea was originally inspired by Jason's Sphereultimate's lovely stickering, in fact.)

You could either do a weave pattern where bands cross, or a bevel-type pattern (ideal for puzzles with intersections of 3+ cuts).

To my knowledge this hasn't been done much. I wish I had lots of spare puzzles to try it on...