  The boundary of a topological insulator (TI) hosts an anomaly restricting its
possible phases: e.g. 3D strong and weak TIs maintain surface conductivity at
any disorder if symmetry is preserved on-average, at least when electron
interactions on the surface are weak. However the interplay of strong
interactions and disorder with the boundary anomaly has not yet been
theoretically addressed. Here we study this combination for the edge of a 2D TI
and the surface of a 3D weak TI, showing how it can lead to an "Anomalous Many
Body Localized" (AMBL) phase that preserves the anomaly. We discuss how the
anomalous Kramers parity switching with pi flux arises in the bosonized theory
of the localized helical state. The anomaly can be probed in localized
boundaries by electrostatically sensing nonlinear hopping transport with e/2
shot noise. Our AMBL construction in 3D weak TIs fails for 3D strong TIs,
suggesting that their anomaly restrictions are distinguished by strong
interactions.
