  We introduce the Dyck path triangulation of the cartesian product of two
simplices $\Delta_{n-1}\times\Delta_{n-1}$. The maximal simplices of this
triangulation are given by Dyck paths, and its construction naturally
generalizes to produce triangulations of $\Delta_{r\ n-1}\times\Delta_{n-1}$
using rational Dyck paths. Our study of the Dyck path triangulation is
motivated by extendability problems of partial triangulations of products of
two simplices. We show that whenever $m\geq k>n$, any triangulation of
$\Delta_{m-1}^{(k-1)}\times\Delta_{n-1}$ extends to a unique triangulation of
$\Delta_{m-1}\times\Delta_{n-1}$. Moreover, with an explicit construction, we
prove that the bound $k>n$ is optimal. We also exhibit interesting
interpretations of our results in the language of tropical oriented matroids,
which are analogous to classical results in oriented matroid theory.
