  We construct the complete coupling of $(1,0)$ supergravity in six dimensions
to $n$ tensor multiplets, extending previous results to all orders in the fermi
fields. We then add couplings to vector multiplets, as dictated by the
generalized Green-Schwarz mechanism. The resulting theory embodies factorized
gauge and supersymmetry anomalies, to be disposed of by fermion loops, and is
determined by corresponding Wess-Zumino consistency conditions, aside from a
quartic coupling for the gaugini. The supersymmetry algebra contains a
corresponding extension that plays a crucial role for the consistency of the
construction. We leave aside gravitational and mixed anomalies, that would only
contribute to higher-derivative couplings.
