We propose a robust and efficient algorithm for the recovery of the joint support in compressed sensing with multiple measurement vectors (the MMV problem). When the unknown matrix of the jointly sparse signals has full rank, MUSIC is a guaranteed algorithm for this problem, achieving the fundamental algebraic bound on the minimum number of measurements. We focus instead on the unfavorable but practically significant case of rank deficiency or bad conditioning. This situation arises with limited number of measurements, or with highly correlated signal components. In this case MUSIC fails, and in practice none of the existing MMV methods can consistently approach the algebraic bounds. We propose subspace-augmented MUSIC, which overcomes these limitations by combining the advantages of both existing methods and MUSIC. It is a computationally efficient algorithm with a performance guarantee.