I invented this version of Heyawake by taking out the outlined rooms from one part of the logic ,while they remain relevant for the other. This example puzzle does not contain many unclued rooms, but I prefer to keep all of the unclued rooms that would show up in this new style, obviously to preserve the rectangular nature, despite their irrelevance to the solve. Now the rules:

Shade in some cells such that all unshaded cells are connected orthogonally. There cannot be more than three uninterrupted unshaded cells in any row or column. Shaded cells may not touch through edges. Numbers indicate how many cells in that room are shaded.