  We show that the following groups are invariably generated; the group of
piecewise projective homeomorphisms of the real line, the group of piecewise
$\mathrm{PSL}(2,\mathbb{Z})$ homeomorphisms of the real line, Monod's group
$H(\mathbb{Z})$, the group of piecewise $\mathrm{PSL}(2,\mathbb{Q})$
homeomorphisms of the real line with rational breakpoints. We also show that
the Higman--Thompson group $F_n$ for every $n \in \mathbb{Z}_{\geq 3}$ and the
golden ratio Thompson group $F_{\tau}$ are invariably generated.
