We consider a specific situation of correlated data where multiple outcomes are repeatedly measured on each member of a couple. Such multivariate longitudinal data from couples may exhibit multi-faceted correlations that can be further complicated if there are polygamous partnerships. An example is data from cohort studies on human papillomavirus (HPV) transmission dynamics in heterosexual couples. HPV is a common sexually transmitted disease with 14 known oncogenic types causing anogenital cancers. The binary outcomes on the multiple types measured in couples over time may introduce inter-type, intra-couple, and temporal correlations. Simple analysis using generalized estimating equations or random effects models lacks interpretability and cannot fully use the available information. We developed a hybrid modeling strategy using Markov transition models together with pairwise composite likelihood for analyzing such data. The method can be used to identify risk factors associated with HPV transmission and persistence, estimate difference in risks between male-to-female and female-to-male HPV transmission, compare type-specific transmission risks within couples, and characterize the inter-type and intra-couple associations. Applying the method to HPV couple data collected in a Ugandan male circumcision (MC) trial, we assessed the effect of MC and the role of gender on risks of HPV transmission and persistence. Supplementary materials for this article are available online.