  For many applications with multivariate data, random field models capturing
departures from Gaussianity within realisations are appropriate. For this
reason, we formulate a new class of multivariate non-Gaussian models based on
systems of stochastic partial differential equations with additive type G noise
whose marginal covariance functions are of Mat\'ern type. We consider four
increasingly flexible constructions of the noise, where the first two are
similar to existing copula-based models. In contrast to these, the latter two
constructions can model non-Gaussian spatial data without replicates.
Computationally efficient methods for likelihood-based parameter estimation and
probabilistic prediction are proposed, and the flexibility of the suggested
models is illustrated by numerical examples and two statistical applications.
