  In recent years, correlated insulating states, unconventional
superconductivity, and topologically non-trivial phases have all been observed
in several moir\'e heterostructures. However, understanding of the physical
mechanisms behind these phenomena is hampered by the lack of local electronic
structure data. Here, we use scanning tunnelling microscopy and spectroscopy to
demonstrate how the interplay between correlation, topology, and local atomic
structure determines the behaviour of electron-doped twisted monolayer-bilayer
graphene. Through gate- and magnetic field-dependent measurements, we observe
local spectroscopic signatures indicating a quantum anomalous Hall insulating
state with a total Chern number of $\pm 2$ at a doping level of three electrons
per moir\'e unit cell. We show that the sign of the Chern number and associated
magnetism can be electrostatically switched only over a limited range of twist
angle and sample hetero-strain values. This results from a competition between
the orbital magnetization of filled bulk bands and chiral edge states, which is
sensitive to strain-induced distortions in the moir\'e superlattice.
