We isolate the purely combinatorial structure possessed by the bases of augmented directed complexes, sufficient to reconstruct the full category of augmented directed complexes with bases. The theory of these structures, which we name Additive Parity Complexes, is an extension of the theory of parity complexes which retains its combinatorial nature, but in which the role of subsets is played instead by multisets. We claim that it is precisely this difference which gives Steiner theory its power.