Initially, I thought you could try a multivariate hypergeometric distribution based test, implemented as MultiHypergeometric in the CRAN package extraDistr.

After searching a while, however, this article also looks very promising:

... developed a theoretical framework for computing the statistical distributions of multi-set intersections based upon combinatorial theory, and then accordingly designed a procedure to efficiently calculate the exact probabilities of multi-set intersections.

Bonus:

We further developed multiple efficient and scalable techniques to visualize multi-set intersections and the corresponding intersection statistics.

The method is available as an R package: SuperExactTest

Hope this helps, -E