  Anomaly mediated supersymmetry breaking implemented in the minimal
supersymmetric standard model (MSSM) is known to suffer from the tachyonic
slepton problem leading to breakdown of electric charge conservation. We show
however that when MSSM is extended to explain small neutrino masses by gauging
the B-L symmetry, the slepton masses can be positive due to the Z' mediation
contributions. We obtain various soft supersymmetry breaking mass spectra,
which are different from those obtained in the conventional anomaly mediation
scenario. Then there would be a distinct signature of this scenario at the LHC.
