Berry curvature is defined as \nabla_{R} \times \vec{A}, where \vec{A} is the Berry connection. Unavoidably, this formula involves partial differentiations of the eigenvectors, often leading to problematic situations where the wavefunction phase is singular (i.e., multivalued). In this case, infinities may appear in the curvature result, which are not connected in any way to degeneracies (i.e., monopoles). Berry, in his seminal paper, showed that there is a way to circumvent this by introducing his extended curvature formula that does not involve any differentiation of the eigenfunctions.
But how is this possible? The initial curvature definition and the extended curvature formula should result in exactly the same Berry field! That said, if there is any phase-related multivaluedness leading to abnormalities in the initial "classic" curvature definition, where do these abnormalities go when transitioning to the extended curvature formula? It should be noted that these abnormalities were truly a physical contribution (with a definite physical meaning) to Berry curvature, and cannot be eliminated in the extended Berry formula.
It really seems that the extended Berry formula (as introduced in Berry's seminal paper) lacks information, as, it produces a different curvature. It is obvious that some erroneous assumptions have been made in the derivation, but what exactly?