Motivated by applications of distributed storage systems to key-value stores, recently, the multi-version coding problem was proposed to store data that is frequently being updated in a distributed storage system. In particular, in multi-version coding, it is desired to store the data consistently, that is, even if all servers do not receive the data updates simultaneously, the decoder can recover the latest possible version of the data. In this paper, we consider the case where there are correlations among various versions of the data. By respectively leveraging update-efficient codes and Slepian-Wolf, we provide two simple multi-version code constructions to show that the storage cost of multi-version codes can be significantly smaller than previous constructions depending on the degree of correlation between the versions. Moreover, we show that our Slepian-Wolf based construction is essentially optimal in a certain correlation regime.