Eigenvalues of monomial matrices

Question

Let $M = PD$, where $P$ is a permutation matrix and $D$ diagonal. If $P$ is also symmetric, then does $M$ have all real eigenvalues?

Answer 1

How about $M = \begin{pmatrix}0&1\\1&0\end{pmatrix}\begin{pmatrix}-1&0\\0&1\end{pmatrix}$?

Answer 2

No. If $P$ is the matrix of a transposition (2 by 2) and $D$ is $diag(1, -1)$ the eigenvlues are imaginary.