The CRISPR (Clustered Regularly Interspaced Short Palindromic Repeats) system is a recently discovered type of adaptive immune defense in bacteria and archaea that functions via directed incorporation of viral and plasmid DNA into host genomes. Here, we introduce a multi-scale model of dynamic coevolution between hosts and viruses in an ecological context that incorporates CRISPR immunity principles. We analyze the model to test whether and how CRISPR immunity induces host and viral diversification and the maintenance of many coexisting strains. We show that hosts and viruses coevolve to form highly diverse communities. We observe the punctuated replacement of extant strains, so that populations have very low similarity compared over the long term. However in the short term, we observe evolutionary dynamics consistent with both incomplete selective sweeps of novel strains and the recurrence of previously rare strains. Coalitions of multiple dominant host strains are predicted to arise because host strains can have nearly identical immune phenotypes mediated by CRISPR defense albeit with different genotypes. We close by discussing how our explicit eco-evolutionary model of CRISPR immunity can help guide efforts to understand the drivers of diversity seen in microbial communities where CRISPR systems are active.