Consider the set $\mathcal{X}_N$ to be the set of $N \times N$ Hermitian matrices whose off-diagonal elements are purely imaginary. Does there exist a unitary $U$ such that matrices of the set $U^*\mathcal{X}_NU$ to be real?
For example, when $N=2$ such a $U$ exists:
$$\begin{bmatrix} 0 & i\\1&0\end{bmatrix}\begin{bmatrix} a & ib\\-ib&c\end{bmatrix}\begin{bmatrix} 0 & 1\\-i&0\end{bmatrix}=\begin{bmatrix} c & b\\b&a\end{bmatrix}$$
Now, what about $N=3$ and higher values of $N$?