We investigate the sample path regularity of multivariate operator-self-similar stable random fields with values in $\mathbb{R}^m$ given by a harmonizable representation. Such fields were introduced in [25] as a generalization of both operator-self-similar stochastic processes and operator scaling random fields and satisfy the scaling property $\{X(c^E t) : t \in \mathbb{R}^d \} \stackrel{\rm d}{=} \{c^D X(t) : t \in \mathbb{R}^d \}$, where $E$ is a real $d \times d$ matrix and $D$ is a real $m \times m$ matrix. This paper provides the first results concerning sample path properties of such fields, including both $E$ and $D$ different from identity matrices. In particular, this solves an open problem in [25].