It would be interesting to know whether the subdivision of $|\Sigma|$ given by the $F\times F^{\vee}$ can be realized as the boundary of a $d$-dimensional polytope (as the picture suggests). The dual of the face lattice would be given by the lattice of intervals in the face lattice of $\Delta$ (or of $\Delta^{\vee}$).

In fact, if there was a realization of the combinatorial type of $\Delta$ with an identification $\mathbb{R}^{d}\cong(\mathbb{R}^{d})^{*}$ such that the normal cones of $F$ and $F^{\vee}$ intersect in their relative interiors, then the Minkowski sum $\Delta+\Delta^{\vee}$ would do the trick. For $d=3$, Koebe’s Theorem [Thu80] guarantees such a realization.