  We examine the transport properties of magnetically doped topological
insulator (TI) thin films subject to correlated non-magnetic disorder. For the
disorder we choose a quasi-periodic potential with a random phase. We restrict
the disorder to a central region, which is coupled to two leads in a clean
quantum spin Hall insulator (QSHI) state and concentrate on different
orientations of the quasi-periodicity in the two-dimensional central region. In
the case of a diagonally oriented or purely longitudinal quasi-periodicity we
find different topological Anderson insulator (TAI) phases, with a quantum
anomalous Hall insulator (QAHI), a quantum spin Chern insulator (QSCI), or a
QSHI phase being realized before the Anderson insulation takes over at large
disorder strength. Quantized transport from extended bulk states is found for
diagonal quasi-periodicity in addition to the above TAI phases that are also
observed for the case of uncorrelated disorder. For a purely transverse
orientation of the quasi-periodicity the emerging QSHI and QSCI phases persist
to arbitrarily strong disorder potential. These topological phase transitions
(except to the Anderson insulator phase), can be understood from a self
consistent Born approximation.
