This paper demonstrates that random, independently chosen equi-dimensional subspaces with a unitarily invariant distribution in a real Hilbert space provide nearly tight, nearly equiangular fusion frames. The angle between a pair of subspaces is measured in terms of the Hilbert-Schmidt inner product of the corresponding orthogonal projections. If the subspaces are selected at random, then a measure concentration argument shows that these inner products concentrate near an average value. Overwhelming success probability for near tightness and equiangularity is guaranteed if the dimension of the subspaces is sufficiently small compared to that of the Hilbert space and if the dimension of the Hilbert space is small compared to the sum of all subspace dimensions.

12 pages AMS LaTeX, no figures