  Let $\Sigma:=\Gamma\backslash\mathbb H$ be a finite volume hyperbolic
surface. We associate to any $\Gamma$-invariant positive crossratio defined on
a dense subset of $\partial\mathbb H$ a canonical geodesic current $\mu$. This
produces a Liouville current $\mu_\rho$ for every maximal framed representation
$\rho:\Gamma\to{\rm Sp}(2n,\mathbf F)$ for a real closed field $\mathbf F$,
which we prove being a weighted multicurve as soon as the field generated by
matrix coefficients of $\rho$ has discrete valuation. When $\mu$ is a measured
lamination, for every $\Gamma$-action on a metric space $X$ admitting a
compatible barycenter, we construct an isometric embedding in $X$ of the
$\mathbf R$-tree $\mathcal T(\mu)$ associated to $\mu$. This applies, for
instance, to the $\rho(\Gamma)$-action on a natural metric space associated to
${\rm Sp}(2n,\mathbf F)$.
