  Let $(M,g)$ be a pseudo-Riemannian manifold of signature $(p,q)$. We compute
the obstruction for a vector bundle $S$ over $(M,g)$ to admit a Dirac operator
whose principal symbol induces on $S$ the structure of a bundle of irreducible
real Clifford modules of complex type, that is, a real spinor bundle of
irreducible complex type. In order to do this, we use the theory of Lipschitz
structures in signature $p-q\equiv_8 3,7$ to reformulate the problem as the
obstruction problem for $(M,g)$ to admit a $\mathrm{Spin}^{o}_{\alpha}$
structure with $\alpha = -1$ if $p-q \equiv_{8} 3$ or $\alpha = +1$ if $ p-q
\equiv_{8} 7$, where
$\mathrm{Spin}^o_+(p,q)=\mathrm{Spin}(p,q)\cdot\mathrm{Pin}_{2,0}$ and
$\mathrm{Spin}^o_-(p,q)=\mathrm{Spin}(p,q)\cdot \mathrm{Pin}_{0,2}$. This
allows computing the obstruction in terms of the Karoubi Stiefel-Whitney
classes of $(M,g)$ and the existence of an auxiliary $\mathrm{O}(2)$ bundle
with prescribed characteristic classes. Furthermore, we explicitly show how a
$\mathrm{Spin}^o_{\alpha}$ structure can be used to construct $S$ and we give
geometric characterizations (in terms of associated bundles) of the conditions
under which the structure group of $S$ reduces to certain natural subgroups of
$\mathrm{Spin}^o_{\alpha}$. Finally, we prove that certain codimension two
submanifolds of spin manifolds and certain products of tori with Grassmanians,
which were not known to admit irreducible real spinor bundles, do admit
$\mathrm{Spin}^{o}_{\alpha}$ structures and therefore do admit real spinor
bundles of irreducible complex type.
