  In this paper, we consider a class of multiterminal source coding problems,
each subject to distortion constraints computed using a specific,
entropy-based, distortion measure. We provide the achievable rate distortion
region for two cases and, in so doing, we demonstrate a relationship between
the lossy multiterminal source coding problems with our specific distortion
measure and (1) the canonical Slepian-Wolf lossless distributed source coding
network, and (2) the Ahlswede-K\"{o}rner-Wyner source coding with side
information problem in which only one of the sources is recovered losslessly.
