On complex matrices with simple spectrum that are unitarily similar to real matrices

Suppose that one should verify whether a given complex $n\times n$ matrix can be converted into a real matrix by a unitary similarity transformation. Sufficient conditions for this property to hold were found in an earlier publication of this author. These conditions are relaxed in the following way: as before, the spectrum is required to be simple, but pairs of complex conjugate eigenvalues $\lambda$, $\bar\lambda$, are now allowed. However, the eigenvectors corresponding to such eigenvalues must not be orthogonal.