On Complex Matrices that Are Unitarily Similar to Real Matrices

There are well-known conditions ensuring that a complex $n\times n$ matrix $A$ can be converted by a similarity transformation into a real matrix. Is it possible to realize this conversion via unitary similarity rather than a general one? The following answer to this question is given in this paper: A matrix $A\in M_n(\mathbb C)$ can be made real by a unitary similarity transformation if and only if $A$ and $\overline A$ are unitarily similar and the matrix $P$ transforming $A$ into $\overline A$ can be chosen unitary and symmetric at the same time. Effective ways for verifying this criterion are discussed.