We address the problem of distributed source coding, i.e. compression of correlated sources that are not co-located and/or cannot communicate with each other to minimize their joint description cost. In this work we tackle the related problem of compressing a source that is correlated with another source which is however available only at the decoder. In contrast to prior information-theoretic approaches, we introduce a new constructive and practical framework for tackling the problem based on the judicious incorporation of channel coding principles into this source coding problem. We dub our approach as DIstributed Source Coding Using Syndromes (DISCUS). We focus in this paper on trellis-structured constructions of the framework to illustrate its utility. Simulation results confirm the power of DISCUS, opening up a new and exciting constructive playing-ground for the distributed source coding problem. For the distributed coding of correlated i.i.d. Gaussian sources that are noisy versions of each other with "correlation-SNR" in the range of 12 to 20 dB, the DISCUS method attains gains of 7-15 dB in SNR over the Shannon-bound using "naive" independent coding of the sources.