In this paper we consider the problem of transmitting spatially Correlated Information Sources (CIS) over the Additive White Gaussian Noise (AWGN) Multiple Access Channel (MAC) with transmitted energy constraint. It is well known that the system performance is optimized if the codewords are designed to take advantage of the correlation among sources in the multiple access channel. To that end, we make use of Rate Compatible Modulation (RCM) codes, whose sparse nature is advantageous to preserve the source correlation in the MAC. In order to exploit the source correlation, the proposed RCM-CIS system is comprised of a set of RCM codes that share the same random structure but with each code having its own weight values that are jointly designed. At the receiver, non-binary decoding is applied to avoid short length cycles that arise in the factor graph, which is obtained by jointly considering the source correlation and the RCM codes. Simulation results show that for high throughput transmission rate RCM-CIS has good performance in terms of the BER vs SNR, attaining values below the Shannon limit if source-channel separation is assumed. As shown by the numerical results, the proposed RCM system inherits the high error floors encountered in point-to-point RCM codes. In order to lower the error floor, we propose the use of a Low Density Generation Matrix (LDGM) code in parallel with the proposed RCM structure. The LDGM system transmits a small fraction of the total coded sequence, and it is capable of correcting the residual errors produced by the RCM-CIS system.