  We show that the bordered-sutured Floer invariant of the complement of a
tangle in an arbitrary 3-manifold $Y$, with minimal conditions on the
bordered-sutured structure, satisfies an unoriented skein exact triangle. This
generalizes a theorem by Manolescu for links in $S^3$. We give a theoretical
proof of this result by adapting holomorphic polygon counts to the
bordered-sutured setting, and also give a combinatorial description of all maps
involved and explicitly compute them. We then show that, for $Y = S^3$, our
exact triangle coincides with Manolescu's. Finally, we provide a graded version
of our result, explaining in detail the grading reduction process involved.
