Apparently, the decomposition of a state into an ensemble of pure states is not unique. I can't understand why, as if I understood correctly a "pure state ensemble decomposition" is just the diagonalization of the density operator

$$\rho=\sum_{k=0}^rp_k|\psi_k\rangle\langle\psi_k| $$

where $r$ is the rank of $\rho$ and $|\psi_k\rangle$ are its eigenvectors with associated eigenvalues $p_k$.

Such a diagonalization is unique up to permutations of the $|\psi_k\rangle$, i.e. there exists a unique basis where $\rho$ is diagonal. How can the pure ensemble decomposition not be unique? What have I misunderstood?