We introduce a new missing-data model, based on a mixture of K Markov processes, and consider the general problem of identifying its parameters. We point out in detail the main difficulties of statistical inference for such models: complete likelihood calculation, parametrization of the stationary distribution and identifiability. We propose a general tractable approach for estimating these models (admitting parametrization of the stationary distribution and identifiability) and check in detail that our assumptions are fully satisfied for a Markov mixture of two linear AR(1) models with Gaussian noise. Finally, a Monte Carlo method is proposed to calculate the split data likelihood of this model when no analytic expression for the invariant probability densities of the Markov processes is known.