As far as I know, when talking about TQFT, one usually means TQFTs on oriented manifolds with boundary (cobordisms)
It appears to me that the Turaev-Viro-Barrett-Westbury state-sum construction can be put on an arbitrary 3-manifold with boundary (cobordism) which needs not be orientable and still yields a valid (unitary) TQFT on those 3-manifolds (cobordisms). The corresponding TQFTs (whose algebraic data is connected to the Drinfeld double of the input fusion category via the Reshetikin-Turaev construction) are said to be non-chiral.
Is that true and is it also the case that (unitary) TQFTs corresponding to modular fusion categories (via Reshetikin-Turaev) that are not Drinfeld doubles (i.e. the chiral ones) cannot be extended to non-orientable manifolds with boundary?