Because of $\mathbb C^n = \mathbb R^{2n}$ complex polytopes ought be embeddable as substructures of real space polytopes (with doubled up dimensionality).
Look, complex edges do have $m\ge2$ vertices and thus are represented as (convex) regular polygon within the Argand plane just as the hull of the roots of unity for example. Similarily any $k$-element of a complex polytope would be contained within a $2k$-dimensional real subspace. As such $n$-dimensional complex polytopes seem possibly be represented as substructures of accordingly chosen $2n$-dimensional polytopes.
Some of these embeddings already are known (cf. eg. Shephard-Coxeter Polytopes. But is there a clear construction on how to re-build those in general, given the (extended) Coxeter-Dynkin diagram of any complex polytope?