Suppose $X$ is a multivariate Gaussian random variable $X\sim \mathcal{N}\left(0,H\right)$ and we define a new random variable $\eta$ by its multiplication with some other random variable $Y$, i.e., $\eta = YX$.

What should I need to require from $Y$ in order that $\eta$ will be Gaussian but will also be uncorrelated with $X$?