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We investigate the use of LDGM codes for the transmission of two highly correlated senders over a Rayleigh fading multiple access channel. Each one of the senders is encoded independently, and without knowledge of the correlation parameters, using an LDGM code. Decoding is performed in a common receiver, which exploits the correlation between the senders by performing message passing in a graph jointly describing the correlation model and the channel codes. The resulting performance surpasses the theoretical limits obtained when separation between source and channel coding is assumed. As opposed to the case of uncorrelated senders, the performance loss when the channel state information is not available at the decoder is practically non-existent.