Any unitary matrix $U$ can be diagonalized by another unitary matrix $V$, $$U=VDV^\dagger,$$ where $D={\rm diag}(z_1,z_2,...,z_N)$ is diagonal.

If $U$ is taken at random uniformly with respect to Haar measure, then $V$ and $D$ are independent and $D$ has the Weyl distribution, $P(D)\propto \prod_{j<k}|z_k-z_j|^2$. I would like to know what is the space of all $V$'s. Which unitary matrices are eigenvectors of unitary matrices? What is their distribution?

On the one hand I would guess that $V$ is also uniformly distributed in the unitary group, but on the other hand this seems paradoxical. Because integration over $U$ can be decomposed as integration over $D$ and $V$ and then integration over $V$ would be the same as integration over $U$ again?

I have consulted many references about this subject, but they tend to focus on the eigenvalues.