Physical manifestation of the Berry connection

Although I have written several papers about it I still struggle to understand the Berry phase and how it is or may be manifested in solids.
Recent reading, summarised below, has helped.

There is a nice short review Geometry and the anomalous Hall effect in ferromagnets
N. P. Ong and Wei-Li Lee

As late as 1999 Sundaram and Niu wrote down the semi-classical equations of motion for Bloch states in the presence of a Berry curvature, script F below.
(1) and (2) below. n.b. how there is a certain symmetry between x and k.

The last equation gives the "magnetic monopoles" associated with the Berry connection/. Aside: the Berry connection Omega_c is the analogue of the magnetic field. It is related to the curvature F tilde by (F tilde)_ab= epsilon_abc Omega_c.

The Berry connection is related to the Berry phase in the same sense that a magnetic field is associated with an Aharonov-Bohm phase.

The above text is taken from a beautiful paper Berry Curvature on the Fermi Surface: Anomalous Hall Effect as a Topological Fermi-Liquid Property by Duncan Haldane.

The symmetry arguments above show why the anomalous Hall effect only occurs in the presence of time-reversal symmetry breaking, e.g. in a ferromagnet.

It is interesting that Robert Karplus (brother of Martin) and Luttinger wrote down what is now called the Berry connection as long ago as 1954! (30 years before Berry!)
They called it the anomalous velocity.
The connection with Berry and topology was only made in 2002 by Jungwirth, Niu, and MacDonald.
An extensive review of the anomalous Hall effect, both theory and experiment, is here.