Filtering and smoothing in switching state-space models are important in numerous applications. The classic family of conditionally Gaussian linear state space models (CGLSSMs) is a natural extension of the Gaussian linear system by introducing its dependence on switches. In spite of their simplicity, recursive filtering and smoothing are no longer feasible in CGLSSMs and approximate methods must be used. Conditionally Markov switching hidden linear models (CMSHLMs) are alternative models which allow recursive optimal exact filtering and smoothing. We introduce an original family of CMSHLMs defined with copulas and we address the problem of their identification. The proposed identification method chooses a model in a family of admissible parametric models and estimates the parameters. It is applied to a learning sample containing observations and states, while the switches are unknown. The interest of the proposed ”semi-unsupervised” filtering and smoothing is validated via experiments on simulated data.