  For a lattice path $\nu$ from the origin to a point $(a,b)$ using steps
$E=(1,0)$ and $N=(0,1)$, we construct an associated flow polytope
$\mathcal{F}_{\hat{G}_B(\nu)}$ arising from an acyclic graph where
bidirectional edges are permitted. We show that the flow polytope
$\mathcal{F}_{\hat{G}_B(\nu)}$ admits a subdivision dual to a $w$-simplex,
where $w$ is the number of valleys in the path $\bar{\nu} = E\nu N$.
Refinements of this subdivision can be obtained by reductions of a polynomial
$P_\nu$ in a generalization of M\'esz\'aros' subdivision algebra for acyclic
root polytopes where negative roots are allowed. Via an integral equivalence
between $\mathcal{F}_{\hat{G}_B(\nu)}$ and the product of simplices
$\Delta_a\times \Delta_b$, we thereby obtain a subdivision algebra for a
product of two simplices. As a special case, we give a reduction order for
reducing $P_\nu$ that yields the cyclic $\nu$-Tamari complex of Ceballos,
Padrol, and Sarmiento.
