To analyze and project age-specific mortality or morbidity rates age-period-cohort (APC) models are very popular. Bayesian approaches facilitate estimation and improve predictions by assigning smoothing priors to age, period and cohort effects. Adjustments for overdispersion are straightforward using additional random effects. When rates are further stratified, for example by countries, multivariate APC models can be used, where differences of stratum-specific effects are interpretable as log relative risks. Here, we incorporate correlated stratum-specific smoothing priors and correlated overdispersion parameters into the multivariate APC model, and use Markov chain Monte Carlo and integrated nested Laplace approximations for inference. Compared to a model without correlation, the new approach may lead to more precise relative risk estimates, as shown in an application to chronic obstructive pulmonary disease mortality in three regions of England & Wales. Furthermore, the imputation of missing data for one particular stratum may be improved, since the new approach takes advantage of the remaining strata if the corresponding observations are available there. This is shown in an application to female mortality in Denmark, Sweden and Norway from the 20th century, where we treat for each country in turn either the first or second half of the observations as missing and then impute the omitted data. The projections are compared to those obtained from a univariate APC model and an extended Lee-Carter demographic forecasting approach using the proper Dawid-Sebastiani scoring rule.