Gaussian marginals do not imply a Gaussian random field

Let $X$ be a standard Gaussian random variable and $S_j$ independent symmetric signs, independent of $X$. The stationary sequence $W_j=S_{j}X$ has standard Gaussian marginals and identity covariance, but is not jointly Gaussian. For two coordinates its moment-generating function is $(1+e^{2t^2})/2$, rather than $e^{t^2}$. Consequently a field's Gaussian one-point densities and two-point covariance alone do not justify an exponential moment of a Gaussian linear functional.