Cache-aided coded multicast leverages side information at wireless edge caches to efficiently serve multiple unicast demands via common multicast transmissions, leading to load reductions that are proportional to the aggregate cache size. However, the increasingly dynamic, unpredictable, and personalized nature of the content that users consume challenges the efficiency of existing caching-based solutions in which only exact content reuse is explored. This paper generalizes the cache-aided coded multicast problem to specifically account for the correlation among content files, such as, for example, the one between updated versions of dynamic data. It is shown that (i) caching content pieces based on their correlation with the rest of the library, and (ii) jointly compressing requested files using cached information as references during delivery, can provide load reductions that go beyond those achieved with existing schemes. This is accomplished via the design of a class of correlation-aware achievable schemes, shown to significantly outperform state-of-the-art correlation-unaware solutions. Our results show that as we move towards real-time and/or personalized media dominated services, where exact cache hits are almost non-existent but updates can exhibit high levels of correlation, network cached information can still be useful as references for network compression.