We give a complete combinatorial characterization of weakly d-Tverberg complexes. These complexes record which intersection combinatorics of convex hulls necessarily arise in any sufficiently large general position point set in Rd. This strengthens the concept of d-representable complexes, which describe intersection combinatorics that arise in at least one point set. Our characterization allows us to construct for every fixed d a graph that is not weakly d '-Tverberg for any d ' <= d, answering a question of De Loera, Hogan, Oliveros, and Yang.
