The theory of hyperplane arrangements is a rich and intensely studied subject, especially from the perspective of combinatorics; see e.g. this wonderful monograph of Stanley. *Oriented* hyperplane arrangements are also a rich subject, leading to the notion of oriented matroid; see e.g. the book of Björner et. al.

I understand that to some extent more general subspace arrangements have been studied, and that for example results about the homological properties of complements of hyperplane arrangements in $\mathbb{C}^n$ can be transferred to this setting. But I wonder whether a theory of *oriented* subspace arrangements has been developed? I would be interested even in the special case in which all subspaces have the same codimension.