Berryʹs phase and Aharonov-Anandanʹs phase

The Aharonov-Anandan geometric phase is generalized to non-unitary evolution, and is shown to be always real. By using a counter-example, which is exactly solvable, it is shown that Berryʹs geometric phase is not always the adiabatic limit of Aharonov-Anandanʹs geometric phase for a non-Hermitian driving Hamiltonion. The reason is then given.