Sign of product of standard normal random variables The question is mainly in the title:

Given two standard normal random variables with correlation $\rho$, what is the distribution of sign of their product?

I understand that when $\rho=0$, we have two iid standard normal random variables and therefore, they take positive and negative values independently with probability $\frac12$.
But I don't know what to do if $\rho\ne0$. We can take $\rho>0$ without loss of generality (because if $X$ and $Y$ are std normal with correlation $\rho$, then $-X$ and $Y$ are std normal with correlation $-\rho$). But I could not proceed further.

Consider $X,Y\sim N(0,1)$ with correlation $\rho$. Then (Nadarajah & Pogány, 2016; Gaunt, 2018) their product is variance-gamma distributed:
$$ XY \sim \text{VG}(1,\rho,\sqrt{1-\rho^2},0). $$
Its PDF is
$$ f_{XY}(x) = \frac{1}{\pi\sqrt{1-\rho^2}}\exp\left(\frac{\rho x}{1-\rho^2}\right) K_0\left(\frac{|x|}{1-\rho^2}\right), $$
where $K_0$ is the modified Bessel function of the second kind of order $0$.
Thus
$$ P(\text{sign}(XY) = -1) = \int_{-\infty}^0 f_{XY}(x)\,dx. $$
You may be able to evaluate this using your favorite computer algebra system. Unfortunately, this exceeds the standard computation time for WolframAlpha. COOLSerdash notes that the integral evaluates to a nice round $\frac{\arccos\rho}{\pi}$.
Alternatively, you could map the above parameterization to the one employed in the VarianceGamma package for R and use the functions in there, if all you are interested in is numerical results.
