Modern coding systems based on optimized low-density parity-check (LDPC) codes now provide near-capacity performance for many channels (e.g., point to point, multiple access, and intersymbol interference). However, these outstanding results usually require LDPC codes that are optimized for particular channel conditions and performance degrades if the true system parameters differ from their assumed values. To highlight this, we call a system universal if it retains near-capacity performance without channel knowledge at the transmitter. Unfortunately, there are a variety of problems (e.g., multiuser communication with fading) where it appears standard systems with optimized LDPC codes cannot be universal. Recently, it has been observed that spatially-coupled LDPC codes appear to be universal across the class of binary memoryless channels. A very interesting open question is whether or not spatially coupling can be extended to provide universality for more general problems. For example, we consider a noisy Slepian-Wolf problem where two (separately encoded) correlated sources are transmitted over two independent erasure channels with unknown erasure rates. The goal is to provide reliable communication, for fixed encoding rates, over the largest possible set of erasure probabilities. Previously, irregular LDPC codes (with punctured systematic bits) were optimized for this problem and the authors were unable to obtain near-universal performance with iterative decoding. This work demonstrates near-universal performance for this problem using a new spatially-coupled coding system. Similar results are also discussed for other problems where iterative information processing with optimized LDPC codes appears unable to provide universality.