  We establish a formula for the L-theory spectrum of real $C^*$-algebras from
which we deduce a presentation of the L-groups in terms of the topological
K-groups, extending all previously known results of this kind. Along the way,
we extend the integral comparison map $\tau\colon \mathrm{k} \to \mathrm{L}$
obtained in previous work by the first two authors to real $C^*$-algebras and
interpret it using topological Grothendieck-Witt theory. Finally, we use our
results to give an integral comparison between the Baum-Connes conjecture and
the L-theoretic Farrell-Jones conjecture, and discuss our comparison map $\tau$
in terms of the signature operator on oriented manifolds.
