  Given a minimal action $\alpha$ of a countable group on the Cantor set, we
show that the alternating full group $\mathsf{A}(\alpha)$ is non-amenable if
and only if the topological full group $\mathsf{F}(\alpha)$ is $C^*$-simple.
This implies, for instance, that the Elek-Monod example of non-amenable
topological full group coming from a Cantor minimal $\mathbb{Z}^2$-system is
$C^*$-simple.
