Abstract We investigate the transmission of multiple correlated binary sources to a single destination over non-orthogonal Gaussian multiple access channels (MACs). By considering a binary codebook, we derive the admissible rate regions of the two-source Gaussian MAC. It is demonstrated that the admissible rate region increases as the correlation between the sources increases. Furthermore, we develop an iterative joint source channel decoding scheme based on systematic irregular low-density parity-check codes by exploiting the correlation between the two sources. The constituent decoders corresponding to each source are implemented in parallel via local iterations, exchanging extrinsic information with each other during the global iterations. Simulation results are provided to verify the performance improvement of the transmission of correlated sources compared to its independent sources counterpart.