  The correlations that violate the CHSH inequality are known to have
complementary contributions from signaling and local indeterminacy. This
complementarity is shown to represent a strengthening of Bell's theorem, and
can be used to certify randomness in a device-independent way, assuming neither
the validity of quantum mechanics nor even no-signaling. We obtain general
nonlocal resources that can simulate the statistics of the singlet state,
encompassing existing results. We prove a conjecture due to Hall (2010) and Kar
et al. (2011) on the complementarity for such resources.
