Engineered complete intersections: eliminating variables and understanding topology

We continue the study of engineered complete intersections (ECI) -- an umbrella generality for a number of important objects in combinatoiral and applied algebraic geometry (such as nondegenerate toric complete intersections, critical loci of their projections, hyperplane arrangements, generalized Calabi--Yau complete intersections, incidence varieties in algebraic optimization, reaction networks). In this paper, we work on extending to ECIs several classical results about toric complete intersections. This includes elimination theory, patchworking over ${\mathbb R}$, and computing basic geometric invariants over ${\mathbb C}$. Our results apply e.g. to eliminating variables in systems of ODEs, such as reaction networks, computing Newton polytopes of discriminants, constructing real polynomial maps and reaction networks with prescribed topology. Along the way, we assign a cohomology ring to an arbitrary tropical fan, and relate reducible ECIs to arrangements of pairwise intersecting planes.