Anyway, here's an idea I got while procrastinating on everything else. This Fillomino variant is based on the snub square tiling, and while you're solving, the triangles count exactly the same as the squares.
More explicitly: write a number in each square or triangular cell so that, starting at any cell and traveling only by crossing edges, only to cells containing the same number, you are able to reach that number of cells total. Alternatively, you can just darken some edges to separate shapes, making sure all pairs of shapes with a dark edge between them have different numbers of cells.
(Any ideas on a better title than Snub Fillomino?)