  The possible existence of a quantum spin liquid (QSL) phase, an exotic state
of matter with long-range quantum entanglement and fractionalized excitations,
in $\alpha$-RuCl$_3$ has sparked widespread interests in exploring QSLs in
various Kitaev models under magnetic fields. Recently, a
$K$-$J$-$\Gamma$-$\Gamma'$ model has been proposed to accurately describe the
compound by fitting the measured thermodynamic data, where $J$ is Heisenberg
interaction, and $\Gamma$, $\Gamma'$ are off-diagonal exchanges on top of the
dominant Kitaev coupling $K$. Based on this effective model, an intermediate
QSL phase in presence of an out-of-plane fields along the $[1 1 1]$ direction,
between the low-field zigzag order and high-field polarized phase, has been
predicted. By combining density matrix renormalization group (DMRG),
exponential tensor renormalization group (XTRG), and variational Monte Carlo
(VMC) calculations, we address the nature of this QSL phase in the honeycomb
$K$-$J$-$\Gamma$-$\Gamma'$ model under the $[1 1 1]$-direction field. Our DMRG
calculations find the algebraic-like decay of spin correlation function, the
finite spin scalar chiral order, and lattice nematic order. Together with the
XTRG results of power-law specific heat at low temperature, our findings
naturally suggest a gapless nematic chiral spin liquid. On the other hand, our
VMC study finds a gapped nematic chiral spin liquid with the variational energy
very close to that obtained in DMRG. As VMC finds a very small gap that is
beyond the current resolution of both ground-state DMRG and finite-temperature
XTRG calculations on finite-width cylinders, we resort the full clarification
for the nature of the intermediate QSL to future studies. Lastly, we discuss
the implications of our results to the recent experiment on $\alpha$-RuCl$_3$
and the QSL-like phase in a similar $K$-$\Gamma$-$\Gamma'$ model.
