Bell inequalities with auxiliary communication

What is the communication cost of simulating the correlations produced by quantum theory? We generalize Bell inequalities to the setting of local realistic theories augmented by a fixed amount of classical communication. Suppose two parties choose one of M two-outcome measurements and exchange 1 bit of information. We present the complete set of inequalities for M = 2, and the complete set of inequalities for the joint correlation observable for M = 3. We find that correlations produced by quantum theory satisfy both of these sets of inequalities. One bit of communication is therefore sufficient to simulate quantum correlations in both of these scenarios.