Let $K_{\alpha}$ be local potentials for a given split Monge-Ampère solution $(V,W)$. We will use the partial Legendre transform to define new coordinates $y_{i},y_{p}$ as in Lemma 4.1. We claim that these are affine coordinates, that is the transition maps are affine linear.

Indeed, by comparing the two local potentials in the overlap $U_{\alpha}\cap U_{\beta}$ we see that $K_{\alpha}-K_{\beta}$ has to be in the form $f(s)g(t)$ where both $f(s)$ and $g(t)$ are affine functions. Thus $y_{\alpha}-y_{\beta}$ are affine functions of $t_{p}$, hence affine functions of $y_{p}$.

Lemma 4.1 also guarantees that $\Psi_{\alpha}$ are local Monge-Ampère potentials in the affine coordinates $y$. The polyhedral property follows from noticing that the charts $U_{v}$ are bounded by linear inequalities in $t_{p}$, and hence by (the same) linear inequalities in the new (affine) coordinates $y_{p}$.

Applying the partial Legendre transform to the other half of the variables $(s,t)$ gives the dual MA structure on $R^{\circ}$. Same considerations as above show that the dual affine structure is polyhedral of type $(\{U_{w}\},\partial\mathcal{T}^{\vee})$.

The final step is to show that the resulting Monge-Ampère bi-PIKAS has the right type, that is, to compute its monodromy around a loop $(v_{i_{1}}w_{j_{1}}v_{i_{2}}w_{j_{2}})$. For this we consider the closed $(N^{*}_{\tau})_{\mathbb{R}}\otimes(N_{\sigma})_{\mathbb{R}}$-valued 1-form on $R^{\circ}$:

Making use of the distributional equation (18) and applying Stokes’ theorem we can compute the integral of $\beta$ along the loop $(v_{i_{1}}w_{j_{1}}v_{i_{2}}w_{j_{2}})=:\partial S$:

Thus, $\beta$ has holonomy which depends only on the homotopy class of the path. That is, we can think of $\beta$ as given by the differential of a multi-valued function

The affine coordinates $y_{p},\ n+1\leq p\leq n+l,$ are single valued. Hence, there is no monodromy in $dy_{p}$. On the other hand, the ambiguity in the remaining differentials

is coming exactly from the multi-valuedness of $f^{iq}$. Thus, the monodromy around the loop $(v_{i_{1}}w_{j_{1}}v_{i_{2}}w_{j_{2}})$ is given by $\mathbbm{1}+2\pi(v_{i_{2}}-v_{i_{1}})\otimes(w_{j_{2}}-w_{j_{1}})$. The factor of $2\pi$ can be absorbed into a redefinition of the affine coordinates.

Tracing backwards through the above argument shows the converse statement is also true. Namely, given a Monge-Ampère bi-PIKAS of type $(\sigma,\tau)$ we can use it together with its dual to define single-valued coordinates $(s,t)$ on $R^{\circ}$ which will obviously extend to $R$. Moreover, due to the polyhedral properties of these bi-PIKAS the discriminant locus in $R$ will be exactly given by $\partial\mathcal{U}_{v}\cap\partial\mathcal{U}_{w}=\Pi(\sigma)\times\Pi(\tau)$, and the prescribed monodromy of the affine structure will guarantee the $(\sigma,\tau)$-type asymptotics of the Monge-Ampère potential $K(s,t)$. ∎