In this letter, distributed source coding with one distortion criterion and correlated messages is considered. This problem can be regarded as “Berger-Yeung problem with correlated messages”. It corresponds to the source coding part of the graph-based framework for transmission of a pair of correlated sources over the multiple-access channel where one is lossless and the other is lossy. As a result, the achievable rate-distortion region for this problem is provided. A rigorous proof of both achievability and converse part is also given.