  The Kitaev model is an exactly solvable quantum spin model within the
language of the constrained real fermions. In spite of numerous studies along
special magnetic-field orientations, there is a limited amount of knowledge on
the complete field-angle characterization, which can provide valuable
information on the existence of fractionalized excitations. For this purpose,
we first extend previous studies on the field-angle response of the
ferromagnetic Kitaev model to its antiferromagnetic version. Yet, the realistic
description of the candidate Kitaev materials, within the edge-sharing
octahedra paradigm, require additional coupling terms, including a large
off-diagonal term $\Gamma$ along with possible anisotropic corrections
$\Gamma_p$. It is therefore not sufficient to depend on the topological
properties of the bare Kitaev model as the only source for the observed thermal
Hall conductivity signals and an understanding of these extended Kitaev models
with a complete field response is demanded. Starting from the zero-field phase
diagram of realistic K-$\Gamma$-$\Gamma_p$ models, we identify
antiferromagnetic zig-zag and (partially) polarized phases as well as two
unusual Kitaev(-$\Gamma$) spin-liquid phases. The magnetic field response of
these phases for arbitrary field orientations provides a remarkably rich phase
diagram. A partially polarized phase is revealed between two ordered phases
with a suppressed magnetization, finite fractionalization and finite Chern
number. This phase is characterized as an extended Kitaev-$\Gamma$ spin-liquid.
To comply our findings with experiments, we reproduce the asymmetry in the
extent of the intermediate phases specifically for the two different field
directions $\theta = \pm 60^o$.
