This variation was introduced by Nikolai Beluhov at Puzzled by Titles, see the authoritative rules and two very tricky puzzles: Fillomino 6: Checkered, Fillomino 8: Checkered. Here’s something that’s hopefully both correct and a little more approachable.

Rules (brief): Solve as a standard Fillomino, and shade some resulting polyominoes, such that no two shaded or unshaded polyominoes share an edge.