Independence of “residuals” in a Bayesian multilevel hierarchical model

So i'm having some problems realising what model checks I should do after fitting a bayesian model other than convergence diagnostics. Lets say i'm fitting a hierarchical bayesian regression model, I take a look at the posterior predictive density $y_{rep}$ is defined as, $$p(y^{rep}|y) = \int(y^{rep}|\theta)p(\theta|y)d\theta$$ Where $\theta$ are the parameters of the model and y is the response. Should I examine if the "residuals": $y-y^{rep}$ are i.i.d? Now as far as I understand why normality is not an issue, how ever isn't independence? In other words should one check if $y-y^{rep}$ is autocorrelated?