Are there any examples of random variables $X_1,\dots,X_n$ over the reals where the marginal distributions $(X_i,X_j)$ for each $i\neq j$ are independent Normal random variables? Of course, setting the $X_i$ to each be independent random Normal variables would be a trivial solution. But are there any nice examples where the $X_i$ are highly dependent?
I'm particularly interested in the case where the vector $X=(X_1,\dots,X_n)$ is constrained to a low-dimensional subspace. Over finite fields, you can set $X=A.Y$ where $A$ is some matrix with $m$ columns and $n$ rows, $m\ll n$, and $Y$ is the uniform distribution over vectors of $m$ field elements. Then as long as the rows of $A$ are pairwise linearly independent, the pairwise marginals of $X$ will be uniformly random in the field. Yet since $m\ll n$, $X$ will be highly non-uniform. What I'm looking for is a version of this over the reals, where the pairwise marginals are Normally distributed instead of uniform.
A non-example: Let $X=(X_1,\dots,X_n)$. For $n$ very large, we can generate a matrix $A$ with $m$ columns and $n$ rows, $m\ll n$, where the rows are all pairwise "almost" orthogonal. Then if we set $X=A.Y$ where $Y$ is just $m$ iid standard Normal variables, then any pair $(X_i,X_j)$ will be ``almost'' uncorrelated Normally distributed variables. Here, "almost" can be made precise. However, since you can only have $m$ of the rows be mutually perfectly orthogonal, you can never get true independence in this way (at least for finite $n$). I'm looking for examples giving true pairwise independent Normal variables for the pairwise marginals.