The extreme conductivity enhancement (by a factor of up to 10(4)) in AgI:Al2O3 composites as well as the anomalous phase transition to the a-phase with a large hysteresis of 50 degrees C was confirmed by conductivity, differential scanning calorimetry, and in situ X-ray diffractometry (XRD). All the composites exhibited an activation energy of 0.29 +/- 0.02 eV in conductivity which was identified with the migration enthalpy for transport in the disordered close-packed layers in AgI system. A close examination of the XRD patterns shows the existence of a seven-layer polytype (7H) for AgI with the stacking sequence ABCBCAC. This may account for anomalies in the composites in contrast to the ideal space charge effect in AgCI:Al2O3 and Ag.Br:Al2O3 composites. Since the stacking arrangement can be regarded as a heterostructure of beta-AgI (ABABAB...) and gamma-AgI (ABCABC...) with a sub-Debye length spacing, the conductivity enhancement may be attributed to the mesoscopic multiphase effect predicted earlier. The hysteresis can be attributed to the elastic effects in the martensitic transformation between alpha-AgI and 7H-AgI.