Pairwise independent gaussians
The posting on MathOverflow tells how to go from a small number of independent Uniform[0,1] random variables to a larger number of pairwise-independent Uniform[0,1] random variables. You can of course go back and forth between Uniform[0,1] and Gaussian by inverting the CDF. But that requires numerical analysis as the CDF is not closed-form.

However, there is a simpler way to from Gaussian to uniform. Given two independent Gaussians $X_1, X_2$, the angle $\arctan(X_1/X_2)$  is uniform in the range $[0,2 \pi]$.

Similarly, the Box-Muller method transforms two independent Uniform[0,1] variables into two independent Gaussian random variables.

Using these two transformations, you consume two Gaussians to produce a uniform or two uniforms to produce a Gaussian. So there is only a factor of $O(1)$ in the sampling efficiency. Furthermore, no inversion of the Normal cdf is required.