Let's think about hypertoric variety, Gale duality.

Given a vector space in $\R^N$, we have $$ V \to \R^N \to (V^\perp)^* $$ $$ V^\perp \to (\R^N)^* \to V^* $$ Great. Given $\eta \in (V^\perp)^*$ and $\xi \in V^*$, we look at the fiber $V_\eta$ and $(V^\perp)_\xi$, they are partitioned by the restriction of the sign partitions in $\R^N$ and $(\R^N)^*$.

Feasible is dual to bounded. So the two sides has the same collection of feasible and bounded.

Consider the Lagrangian $(V \oplus V^\perp)_{\eta, \xi}$ intersecting with the diagonal sign blocks in $T^*\R^N$.

I still don't know why we have this correspondence of bounded and feasible chambers. But we do, and the two linear spaces $V$ and $V^\perp$ can be really different.