  This paper develops a Cambrian extension of the work of C. Ceballos, A.
Padrol and C. Sarmiento on $\nu$-Tamari lattices and their tropical
realizations. For any signature $\varepsilon \in \{\pm\}^n$, we consider a
family of $\varepsilon$-trees in bijection with the triangulations of the
$\varepsilon$-polygon. These $\varepsilon$-trees define a flag regular
triangulation $\mathcal{T}^\varepsilon$ of the subpolytope $\operatorname{conv}
\{(\mathbf{e}_{i_\bullet}, \mathbf{e}_{j_\circ}) \, | \, 0 \le i_\bullet <
j_\circ \le n+1 \}$ of the product of simplices $\triangle_{\{0_\bullet, \dots,
n_\bullet\}} \times \triangle_{\{1_\circ, \dots, (n+1)_\circ\}}$. The oriented
dual graph of the triangulation $\mathcal{T}^\varepsilon$ is the Hasse diagram
of the (type $A$) $\varepsilon$-Cambrian lattice of N. Reading. For any
$I_\bullet \subseteq \{0_\bullet, \dots, n_\bullet\}$ and $J_\circ \subseteq
\{1_\circ, \dots, (n+1)_\circ\}$, we consider the restriction
$\mathcal{T}^\varepsilon_{I_\bullet, J_\circ}$ of the triangulation
$\mathcal{T}^\varepsilon$ to the face $\triangle_{I_\bullet} \times
\triangle_{J_\circ}$. Its dual graph is naturally interpreted as the increasing
flip graph on certain $(\varepsilon, I_\bullet, J_\circ)$-trees, which is shown
to be a lattice generalizing in particular the $\nu$-Tamari lattices in the
Cambrian setting. Finally, we present an alternative geometric realization of
$\mathcal{T}^\varepsilon_{I_\bullet, J_\circ}$ as a polyhedral complex induced
by a tropical hyperplane arrangement.
