In this answer, it was shown that the product of two i.i.d. standard normal r.v.s is not normally distributed. What if we scale one of them? In other words, if $X, Y \text{ i.i.d } \sim N(0, 1)$, what is the distribution of $Z := X\frac{Y}{|Y|}$? By looking at he distribution function of $Z$ (though I didn't finish the calculation), $Z$ is unlikely to be normal. However, the simulation seems to suggest that $Z$ is normal (the QQ plot of $10,000$ realizations of $Z$ is shown below). Is there a clever way to prove or disprove $Z$ is normally distributed?
Conditional on $Y = y$, $Z$ is $\mathcal N(0,1)$. Since the regular conditional distribution of $Z$ is the same for all $y$, we conclude $Z$ is independent of $Y$. In particular, $Z\sim \mathcal N(0,1)$.
Generalizing, if $X\sim \mathcal N(\mu,\sigma^2_1),Y\sim \mathcal N(0,\sigma^2_2)$ one still has $Z\sim \mathcal N(0,\sigma_1^2)$. It is easy to see that this fails if $E(Y)\neq 0$.