Let A be a Matrix with no real eigenvalues. It is possible that the following Matrix B has real eigenvalues?

B=A-A*A^{T}+A^{T}

If I multiply a Matrix with it's transponse I get a symmetric matrix, which has two real roots so A*A^{T}has real eigenvalues. If I subtract A and add A^{T}it is not symmetric anymore... but how can I derive if it still has no real eigenvalues?