Method to use mvnrnd with very large covariance matrices?

I need to generate 10,000 binary vectors, between each having a specified correlation coefficient (these are representing spike trains). I am attempting to use the mvnrnd function to use it (by first generating correlated random number streams and then converting to binary). It works great for up to around 200 vectors (for the vector length that I am using), but the function needs too much memory above this number and ceases to work. I want to get to 10,000 of these correlated vectors.

Is there any way in MATLAB to get around this problem in order to achieve this goal? (obviously outside of getting more RAM).

Thanks a lot.