I think the difficulty is worse than just finding the right algorithm. The first matter of business is deciding *which* conditional distribution you want to draw from, because they are non-unique.

I'm going to ignore the restricted support, which is a red herring. Instead, consider a bivariate normal distribution with correlation $\rho$: $$(X_1, X_2)^t \sim \mbox{N}((0,0)^t,\Sigma_{\rho}).$$

Now suppose we want to sample from this distribution, restricted to the line $X_1 = X_2$. The measure on this set is non-unique. It will be different depending on if you condition on $Z = 1$, with $Z = X_1/X_2$ versus $Z=0$, with $Z = X_1 - X_2$. I asked about this in my question here (and provide some references, unfortunately behind a pay-wall).

You start out with independent Gaussian random variables, but one you restrict to some rectangle the random variables are no longer independent and the issue above arises. I think it is an interesting question as to how Gibbs samplers relate to particular choices of conditioning variable ($Z$ in my example). For example, you can imagine drawing $X_1$ from its marginal distribution on your rectangle and then just setting $X_2$ to whatever value you have drawn. But this is obviously different than drawing $X_2$ from its marginal distribution and setting $X_1$ to whatever value you have drawn. I don't have a good sense for how these two methods relate to the formal definition of a conditioning sigma algebra. It would be useful to know about because it is often easier to think in terms of Gibbs samplers than it is to think about formal limits.