The joint design of both transmit constellation and low-density parity-check codes (LDPC) for the two-user, symbol-synchronous, binary-input Gaussian multiple access channel is considered. A transmission scheme is proposed to approach the symmetric capacity without the use of time-sharing or rate-splitting by joint decoding of the noisy sum of two LDPC codewords. This scheme relies on an extension of the classic belief propagation (BP) algorithm which allows for the simultaneous decoding of two LDPC codewords. We use a Gaussian approximation (GA) of the message distribution to investigate the convergence of the decoding process and derive a linear programming technique for joint code design. We also implement a superposition modulation scheme to achieve higher rate. This code design is applied to different input constellation choices which attain the symmetric capacity in different SNR regimes. It is shown that, quite surprisingly, in the moderate SNR regime the best performance is obtained by an asymmetric constellation.