  Let $G$ be a compact, connected, and simply-connected Lie group viewed as a
$G$-space via the conjugation action. The Freed-Hopkins-Teleman Theorem (FHT)
asserts a canonical link between the equivariant twisted $K$-homology of $G$
and its Verlinde algebra. In this paper we give a generalization of FHT in the
presence of a Real structure of $G$. Along the way we develop preliminary
materials necessary for this generalization, which are of independent interest
in their own right. These include the definitions of Real Dixmier-Douady
bundles, the Real third cohomology group which is shown to classify the former,
and Real $\text{Spin}^c$ structures.
