  Given a lattice path $\nu$, the alt $\nu$-Tamari lattice is a partial order
recently introduced by Ceballos and Chenevi\`ere, which generalizes the
$\nu$-Tamari lattice and the $\nu$-Dyck lattice. All these posets are defined
on the set of lattice paths that lie weakly above $\nu$, and posses a rich
combinatorial structure. In this paper, we study the geometric structure of
these posets. We show that their Hasse diagram is the edge graph of a polytopal
complex induced by a tropical hyperplane arrangement, which we call the alt
$\nu$-associahedron. This generalizes the realization of $\nu$-associahedra by
Ceballos, Padrol and Sarmiento. Our approach leads to an elegant construction,
in terms of areas below lattice paths, which we call the canonical realization.
Surprisingly, in the case of the classical associahedron, our canonical
realization magically recovers Loday's ubiquitous realization, via a simple
affine transformation.
