The Reshetikhin-Turaev construction takes a modular tensor category $\mathcal C$ and produces a 3-2-1 oriented TQFT $Z_{\mathcal C}$ such that $Z_{\mathcal C}(S^1) = \mathcal C$.
Is there an analogous construction for 3-2-1 unoriented TQFTs? What kind of structure occurs on $Z(S^1)$ in this case?
For context, I have an unextended 3-dimensional TQFT of unoriented manifolds which I'd like to extend to the circle. Its restriction to oriented manifolds isn't very interesting, so I'm hoping for additional structure on $Z(S^1)$ to capture the unorientable part.
(Turaev and Turner answer the analogous question in dimension 2, defining “extended Frobenius algebras” that are equivalent to unoriented 2D TQFTs.)
