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We address the problem of compressing correlated distributed sources, i.e., correlated sources which are not co-located or which cannot cooperate to directly exploit their correlation. We consider the related problem of compressing a source which is correlated with another source that is available only at the decoder. This problem has been studied in the information theory literature under the name of the Slepian-Wolf (1973) source coding problem for the lossless coding case, and as "rate-distortion with side information" for the lossy coding case. We provide a constructive practical framework based on algebraic trellis codes dubbed as DIstributed Source Coding Using Syndromes (DISCUS), that can be applicable in a variety of settings. Simulation results are presented for source coding of independent and identically distributed (i.i.d.) Gaussian sources with side information available at the decoder in the form of a noisy version of the source to be coded. Our results reveal the promise of this approach: using trellis-based quantization and coset construction, the performance of the proposed approach is 2-5 dB from the Wyner-Ziv (1976) bound.