Given two Gaussian random variables $Z:=(Z_1,Z_2)\sim N(0,\Sigma)$, where $\Sigma\in\mathbb{R}^{2\times 2}$, is there an explicit expression for the mean and variance of $Z$ conditioned on $Z_1>Z_2$? Based on a previous question and answer, I know they are not jointly Gaussian. We can assume independence between $Z_1$ and $Z_2$ if it makes things considerably easier.
For example, using $\Sigma=[[1, 0.5],[0.5, 1]]$ and plotting the histograms of the conditioned $Z_1$ and $Z_2$, they look marginally Gaussian to me with $\mathbb{E}[Z_1|Z_1>Z_2]$ being slightly above zero and $\mathbb{E}[Z_2|Z_1>Z_2]$ being slightly below zero.
Picture of $Z_1|Z_1>Z_2$:
Picture of $Z_2|Z_1>Z_2$:
First, recall that if $X\sim N(\mu_X,\sigma^2_X),Y\sim N(\mu_Y,\sigma^2_Y)$ are bivariate normal with correlation $\rho_{X,Y}$,
$$X|Y=y\quad \sim N\left(\mu_X+\rho_{X,Y} {\sigma_X \over \sigma_Y}(y-\mu_Y),(1-\rho_{X,Y}^2)\sigma^2_X\right).$$
Also, recall from the truncated normal distribution that if $X\sim N(\mu,\sigma^2)$
$$E[X \mid X>a]=\mu+\sigma{\phi(\alpha)/Q}\\ \text{Var}(X \mid X>a)=\sigma^2\left(1+\alpha\phi(\alpha)/Q-(\phi(\alpha)/Q)^2 \right)\\ \alpha:={a-\mu \over \sigma}, Q:=1-\Phi(\alpha),$$
where $\phi,\Phi$ are the pdf and cdf respectively of a standard normal. Note $\phi(\alpha)/Q$ is the inverse Mills ratio.
Now define $W:=Z_1-Z_2.$ Observe $$(Z_1,W)':=AZ'\sim N(0,A\Sigma A'),\\(Z_2,W)':=BZ'\sim N(0,B\Sigma B'),$$
$$A:=\left(\begin{array}{cc} 1 & 0\\ 1 & -1 \end{array}\right), B:=\left(\begin{array}{cc} 0 & 1\\ 1 & -1 \end{array}\right).$$
Let's define $\sigma_{Z_i}^2:=\text{Var}(Z_i),\rho:=\text{Corr}(Z_1,Z_2)$, and note that
$$\sigma^2_W:=\text{Var}(W)=\sigma^2_{Z_1}+\sigma^2_{Z_2}-2\rho\sigma_{Z_1}\sigma_{Z_2}\\ \alpha:=\text{Corr}(Z_1,W)={\sigma_{Z_1}-\rho\sigma_{Z_2} \over \sigma_{W}}\\ \beta:=\text{Corr}(Z_2,W)={\rho\sigma_{Z_1}-\sigma_{Z_2} \over \sigma_{W}}.\\ $$
Next observe by the laws of Adam and Eve and above results, $$E[Z_1 \mid W>0]=E[E[Z_1 \mid W] \mid W>0]=E\left[\alpha{\sigma_{Z_1} \over \sigma_W}W \mid W>0\right]=\alpha\sigma_{Z_1} \sqrt{2 \over \pi}\\ \begin{align}\text{Var}(Z_1 \mid W>0)&=\text{Var}(E[Z_1|W] \mid W>0)+E[\text{Var}(Z_1|W)|W>0]\\ &=\text{Var}\left(\alpha{\sigma_{Z_1} \over \sigma_W}W \mid W>0\right)+(1-\alpha^2)\sigma^2_{Z_1}\\ &=\alpha^2\sigma^2_{Z_1}\left(1-2/\pi\right)+(1-\alpha^2)\sigma^2_{Z_1}.\\ \end{align} $$ Likewise,
$$E[Z_2 \mid W>0]=\beta \sigma_{Z_2}\sqrt{ 2 \over \pi}\\ \text{Var}(Z_2 \mid W>0)=\beta^2\sigma^2_{Z_2}\left(1-2/\pi\right)+(1-\beta^2)\sigma^2_{Z_2}.$$