No, unfortunately not. At the moment stan_mvmer only allows for group-level parameters that are correlated across submodels, but not that are common or shared across submodels.
In any case, only the group-level parameters are correlated in stan_mvmer, and not the population-level parameters or residual errors. Hence, I chose to call it stan_mvmer, and not stan_mvglm. I think there is probably a demand for some type of stan_mvglm function that works with correlated errors or multivariate outcome distributions or something. But it just wasn’t relevant to the motivating application behind stan_mvmer.