Can the joint distribution of N normally distributed random variables not be the multivariant normal distribution?

Question

Given a random vector of size $n$, where the marginal distribution of each component $x_i$ is a normal distribution: $x_i \sim N(\mu_i,\sigma_i)$, is it possible that the joint distribution is not the multivariate normal distribution $N(\mu,\Sigma)$? I believe it is possible but cannot come up with an example.

Answer 1

Yes it is possible :

$X \sim N(0,1)$, $U$ a Rademacher random variable independent from $X$.

Then we take $Z = (X,Y) = (X,UX)$

You can show that $Y \sim N(0,1)$ and $P(X=Y) = 1/2$

Therefore, $Z$ is not a continuous random variable.