  Let $G$ be a connected, linear, real reductive Lie group with compact centre.
Let $K<G$ be compact. Under a condition on $K$, which holds in particular if
$K$ is maximal compact, we give a geometric expression for the multiplicities
of the $K$-types of any tempered representation (in fact, any standard
representation) $\pi$ of $G$. This expression is in the spirit of Kirillov's
orbit method and the quantisation commutes with reduction principle. It is
based on the geometric realisation of $\pi|_K$ obtained in an earlier paper.
This expression was obtained for the discrete series by Paradan, and for
tempered representations with regular parameters by Duflo and Vergne. We obtain
consequences for the support of the multiplicity function, and a criterion for
multiplicity-free restrictions that applies to general admissible
representations. As examples, we show that admissible representations of
$\mathrm{SU}(p,1)$, $\mathrm{SO}_0(p,1)$ and $\mathrm{SO}_0(2,2)$ restrict
multiplicity-freely to maximal compact subgroups.
