  Let $LG$ be the loop group of a compact, connected Lie group $G$. We show
that the tangent bundle of any proper Hamiltonian $LG$-space $\mathcal{M}$ has
a natural completion $\overline{T}\mathcal{M}$ to a strongly symplectic
$LG$-equivariant vector bundle. This bundle admits an invariant compatible
complex structure within a natural polarization class, defining an
$LG$-equivariant spinor bundle $\mathsf{S}_{\overline{T}\mathcal{M}}$, which
one may regard as the Spin$_c$-structure of $\mathcal{M}$. We describe two
procedures for obtaining a finite-dimensional version of this spinor module. In
one approach, we construct from $\mathsf{S}_{\overline{T}\mathcal{M}}$ a
twisted Spin$_c$-structure for the quasi-Hamiltonian $G$-space associated to
$\mathcal{M}$. In the second approach, we describe an `abelianization
procedure', passing to a finite-dimensional $T\subset LG$-invariant submanifold
of $\mathcal{M}$, and we show how to construct an equivariant
Spin$_c$-structure on that submanifold.
