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.
Maybe Puzzles 
A lovely layout and a delightful solve, thanks :)
(And I’m pretty certain that it is indeed correct.)
Really nice puzzle. And thanks for making it paint friendly. I can basically get through this variant completely using flood fill.