  We show that the Hamiltonians of the open relativistic Toda system are
elements of the generic basis of a cluster algebra, and in particular are
cluster characters of nonrigid representations of a quiver with potential.
Using cluster coordinates defined via spectral networks, we identify the phase
space of this system with the wild character variety related to the periodic
nonrelativistic Toda system by the wild nonabelian Hodge correspondence. We
show that this identification takes the relativistic Toda Hamiltonians to
traces of holonomies around a simple closed curve. In particular, this provides
nontrivial examples of cluster coordinates on $SL_n$-character varieties for $n
> 2$ where canonical functions associated to simple closed curves can be
computed in terms of quivers with potential, extending known results in the
$SL_2$ case.
