Great question and unfortunately not an uncommon situation. One solution outside of data-analysis is to pool data with other institutions, a solution which has its own obstacles.

Unfortunately it's trail-and-error: i set a threshold below which values are turned to 0, then i invert the matrix back, recalculate the correlation matrix and check if the differences are acceptable. This ensures a PD matrix, but finds less conditional independencies than better algorithms. I whish i understood the stability of matrix inversion better, i asked a related question about that: stats.stackexchange.com/questions/145412/…

@whuber ah, i see what you mean.