There are things that can be done, but almost nothing besides the constructions that Stan uses guarantees positive definiteness. One thing is to utilize the fact that a convex combination of correlation matrices is a correlation matrix. So, if you have \mathbf{\Lambda}_0 as the common correlation matrix and \mathbf{\Lambda}_j as the deviation in the correlation matrix for the j-th group, then \left(1 - \alpha\right) \mathbf{\Lambda}_0 + \alpha \mathbf{\Lambda}_j is the correlation matrix for the j-th group with \alpha \in \left[0,1\right]. You could then put LKJ priors on \mathbf{\Lambda}_j with shape \frac{1}{\alpha} or e^\alpha to encourage the group-specific deviations to be small.