  We develop the theory of spinorial polyforms associated with bundles of
irreducible Clifford modules of non-simple real type, obtaining a precise
characterization of the square of an irreducible real spinor in signature
$(p-q) = 1\,\mathrm{mod(8)}$ as a polyform belonging to a semi-algebraic real
set. We use this formalism to study differential spinors on Lorentzian
three-manifolds, proving that in this dimension and signature, every
differential spinor is equivalent to an isotropic line preserved in a given
direction by a metric connection with prescribed torsion. We apply this result
to investigate Lorentzian three-manifolds equipped with a skew-torsion parallel
spinor, namely a spinor parallel with respect to a metric connection with
totally skew-symmetric torsion. We obtain several structural results about this
class of Lorentzian three-manifolds, which are necessarily Kundt, and in the
compact case, we obtain an explicit differential condition that guarantees
geodesic completeness. We further elaborate on these results to study the
supersymmetric solutions of three-dimensional NS-NS supergravity, which involve
skew-torsion parallel spinors whose torsion is given by the curvature of a
curving on an abelian bundle gerbe. In particular, we obtain a correspondence
between NS-NS supersymmetric solutions and null coframes satisfying an explicit
exterior differential system that we solve locally.
