This paper focuses on the data fusion scenario where N nodes sense and transmit the data generated by a source S to a common destination, which estimates the original information from S more accurately than in the case of a single sensor. This work joins the upsurge of research interest in this topic by addressing the setup where the sensed information is transmitted over a Gaussian Multiple-Access Channel (MAC). We use Low Density Generator Matrix (LDGM) codes in order to keep the correlation between the transmitted codewords, which leads to an improved received Signal-to-Noise Ratio (SNR) thanks to the constructive signal addition at the receiver front-end. At reception, we propose a joint decoder and estimator that exchanges soft information between the N LDGM decoders and a data fusion stage. An error-correcting Bose, Ray-Chaudhuri, Hocquenghem (BCH) code is further applied suppress the error floor derived from the ambiguity of the MAC channel when dealing with correlated sources. Simulation results are presented for several values of N and diverse LDGM and BCH codes, based on which we conclude that the proposed scheme outperforms significantly (by up to 6.3 dB) the suboptimum limit assuming separation between Slepian-Wolf source coding and capacity-achieving channel coding.