  In this work, we propose an exactly solvable two-dimensional lattice model of
strongly correlated electrons that realizes a quantum anomalous Hall insulator
with Chern number $\mathcal{C}=1$. First, we show that the interplay of ionic
potential, Rashba spin-orbit coupling and Zeeman splitting leads to the
appearance of quantum anomalous Hall effect. Next, we calculate in an exact
manner Chern number for the correlated system where electron-electron
interactions are introduced in the spirit of Hatsugai-Kohmoto model using two
complementary methods, one relying on the properties of many-body groundstate
and the other utilizing single-particle Green's function, and subsequently we
determine stability regions. By leveraging the presence of inversion symmetry
we find boundaries between topological and trivial phases on the analytical
ground. Notably, we show that in the presence of correlations onset of
topological phase is no longer signalled by a spectral gap closing consistently
with phenomenon called in literature as {\it first-order topological
transition}. We provide a clear microscopic understanding of this inherently
many-body feature by pinpointing that the lowest energy excited states in the
correlated system are no longer of the single-particle nature and thus are not
captured by a spectral function.
