  In the pseudospin-$\frac{1}{2}$ honeycomb Mott insulators with strong
spin-orbit coupling, there are two types of bond-dependent exchange
interactions, named Kitaev ($K$) and $\Gamma$, leading to strong frustration.
While the ground state of the Kitaev model is a quantum spin liquid with
fractionalized excitations, the ground state of the $\Gamma$ model remains
controversial. In particular, the phase diagram of the $K\Gamma$ model with
ferromagnetic $K$ and antiferromagnetic $\Gamma$ interactions has been
intensively studied because of its relevance to candidate materials such as
$\alpha$-RuCl$_{3}$. Numerical studies also included the effects of tuning the
bond strengths, i.e., $z$-bond strength different from the other bonds.
However, no clear consensus on the overall phase diagram has been reached yet.
Here we study the classical $K\Gamma$ model with the anisotropic bond strength
using Monte Carlo simulations to understand the emerging phases out of
competition between the two frustrated limits. We also address how the
anisotropic bond strength affects the phase diagram and strength of quantum
fluctuations. We found various large unit cell phases due to the competing
frustrations, and analyzed their intrinsic degeneracy based on the symmetry of
the Hamiltonian. Using the linear spin wave theory we showed that the
anisotropic bond strength enhances quantum fluctuations in the
$\Gamma$-dominant regime where a small reduced moment is observed. The
implications of our findings in relation to the quantum model are also
discussed.
