Consider a product of two locally-finite, infinite, unimodular trees $X=T_1\times T_2$. Assume that both ${\rm Aut}(T_1)$ and ${\rm Aut}(T_2)$ are not discrete.

So as a vague general question, what is known about non-uniform lattices in ${\rm Isom}(X)$?

Clearly, if one of the $T_i$s admit a non-uniform lattice $\Gamma_1$ (say $T_1$) then we can take any lattice $\Gamma_2$ in $T_2$ and the direct product $\Gamma=\Gamma_1\times\Gamma_2$ will be a non-uniform lattice in ${\rm Isom}(X)$. However, this lattice is reducible (both as a lattice or an abstract group).

A more specific question is as follows. What are examples of non-uniform lattices with non-discrete projection to $T_1$ and $T_2$? What is known about them (e.g. density of the commensurators, residual finiteness etc.)?