  We study Haar unitary random matrices with permuted entries. For a sequence
of permutations $\left(\sigma_N\right)_N$, where $\sigma_N$ acts on $N\times N$
matrices we identify conditions under which the $\ast$--distribution of
permuted Haar unitary matrices $U_N^{\sigma_N}$ is asymptotically circular and
free from the unpermuted sequence $U_N$. We show that this convergence takes
place in the almost sure sense. Moreover we show that our conditions on the
sequence of permutations are generic in the sense that are almost surely
satisfied by a sequence of random permutations.
