Let $X_0,X_1$ be random variables such that $E[X_1|X_0]=X_0$, $X_0 \sim \mathcal N (0, \sigma_0^2)$, and $X_1 \sim \mathcal N (0, \sigma_1^2)$ (notice that $\sigma_0^2 \leq \sigma_1^2$ by the martingale condition).
I am wondering if the joint distribution of $(X_0,X_1)$ can be non-Gaussian under the above conditions (without the martingale condition, it is easy). And if so, could I see an example?
A Gaussian example (and it is unique) is given by the conditional distribution $X_1|X_0 \sim \mathcal N (X_0, \sigma_1^2 -\sigma_0^2)$