  The eigenvalues of the Hatano--Nelson non-Hermitian Anderson matrices, in the
spectral regions in which the Lyapunov exponent exceeds the non-Hermiticity
parameter, are shown to be real and exponentially close to the Hermitian
eigenvalues. This complements previous results, according to which the
eigenvalues in the spectral regions in which the non-Hermiticity parameter
exceeds the Lyapunov exponent are aligned on curves in the complex plane.
