This paper considers the problem of communicating correlated information from multiple source nodes over a network of noiseless channels to multiple destination nodes, where each destination node wants to recover all sources. The problem involves a joint consideration of distributed compression and network information relaying. Although the optimal rate region has been theoretically characterized, it was not clear how to design practical communication schemes with low complexity. This work provides a partial solution to this problem by proposing a low-complexity scheme for the special case with two sources whose correlation is characterized by a binary symmetric channel. Our scheme is based on a careful combination of linear syndrome-based Slepian-Wolf coding and random linear mixing (network coding). It is in general suboptimal; however, its low complexity and robustness to network dynamics make it suitable for practical implementation.