  In this paper, we exploit the combinatorics and geometry of triangulations of
products of simplices to derive new results in the context of Catalan
combinatorics of $\nu$-Tamari lattices. In our framework, the main role of
"Catalan objects" is played by $(I,\overline{J})$-trees: bipartite trees
associated to a pair $(I,\overline{J})$ of finite index sets that stand in
simple bijection with lattice paths weakly above a lattice path
$\nu=\nu(I,\overline{J})$. Such trees label the maximal simplices of a
triangulation whose dual polyhedral complex gives a geometric realization of
the $\nu$-Tamari lattice introduced by Pr\'evile-Ratelle and Viennot. In
particular, we obtain geometric realizations of $m$-Tamari lattices as
polyhedral subdivisions of associahedra induced by an arrangement of tropical
hyperplanes, giving a positive answer to an open question of F.~Bergeron.
  The simplicial complex underlying our triangulation endows the $\nu$-Tamari
lattice with a full simplicial complex structure. It is a natural
generalization of the classical simplicial associahedron, alternative to the
rational associahedron of Armstrong, Rhoades and Williams, whose $h$-vector
entries are given by a suitable generalization of the Narayana numbers.
  Our methods are amenable to cyclic symmetry, which we use to present type $B$
analogues of our constructions. Notably, we define a partial order that
generalizes the type $B$ Tamari lattice, introduced independently by Thomas and
Reading, along with corresponding geometric realizations.
