Variational inference is routinely deployed in Bayesian state-space models as an efficient computational technique. Motivated by the inconsistency issue observed by Wang and Titterington (2004) for the mean-field approximation in linear state-space models, we consider a more expressive variational family for approximating the joint posterior of the latent variables to retain their dependence, while maintaining the mean-field (i.e. independence) structure between latent variables and parameters. In state-space models, such a latent structure adapted mean-field approximation can be efficiently computed using the belief propagation algorithm. Theoretically, we show that this adapted mean-field approximation achieves consistency of the variational estimates. Furthermore, we derive a non-asymptotic risk bound for an averaged alpha-divergence from the true data generating model, suggesting that the posterior mean of the best variational approximation for the static parameters shows optimal concentration. From a broader perspective, we add to the growing literature on statistical accuracy of variational approximations by allowing dependence between the latent variables, and the techniques developed here should be useful in related contexts.