  Let $(Y_k)_{k\in \mathbb{Z}}$ be a stationary sequence on a probability space
$(\Omega,\mathcal{A},\mathbb{P})$ taking values in a standard Borel space
$\mathsf{Y}$. Consider the associated maximum likelihood estimator with respect
to a parametrized family of hidden Markov models such that the law of the
observations $(Y_k)_{k\in \mathbb{Z}}$ is not assumed to be described by any of
the hidden Markov models of this family. In this paper we investigate the
consistency of this estimator in such misspecified models under mild
assumptions.
