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The most commonly used version of a linear inverse model (LIM) is forced by state-independent noise. Although having several desirable qualities, this formulation can only generate long-term Gaussian statistics. LIM-like systems forced by correlated additive-multiplicative (CAM) noise have been shown to generate deviations from Gaussianity, but parameter estimation methods are only known in the univariate case, limiting their use for the study of coupled variability. This paper presents a methodology to calculate the parameters of the simplest multivariate LIM extension that can generate long-term deviations from Gaussianity. This model (CAM-LIM) consists of a linear deterministic part forced by a diagonal CAM noise formulation, plus an independent additive noise term. This allows for the possibility of representing asymmetric distributions with heavier-or lighter-than-Gaussian tails. The usefulness of this methodology is illustrated in a locally coupled two-variable ocean-atmosphere model of midlatitude variability. Here, a CAM-LIM is calculated from ocean weather station data. Although the time-resolved dynamics is very close to linear at a time scale of a couple of days, significant deviations from Gaussianity are found. In particular, individual probability density functions are skewed with both heavy and light tails. It is shown that these deviations from Gaussianity are well accounted for by the CAM-LIM formulation, without invoking nonlinearity in the time-resolved operator. Estimation methods using knowledge of the CAM-LIM statistical constraints provide robust estimation of the parameters with data lengths typical of geophysical time series, for example, 31 winters for the ocean weather station here.