  Frustrated magnets with highly degenerate ground states are at the heart of
hunting exotic states of matter. Recent studies in spin-orbit coupled honeycomb
magnets have generated immense interest in bond-dependent interactions,
appreciating a symmetric off-diagonal $\Gamma$ interaction which exhibits a
macroscopic degeneracy in the classical limit. Here, we study a generic spin
model and discover a novel chiral-spin ordering with spontaneously broken
time-reversal symmetry near the dominant $\Gamma$ region. The chiral-spin phase
is demonstrated to possess a staggered chirality relation in different
sublattices, and it exhibits gapless excitations as revealed by the vanishing
energy gap and the finite central charge on cylinders. Although there is a
vestige of a tiny peak in the corner of the second Brillouin zone, the magnetic
order is likely to vanish as the system size increases. Finally, we also
attempt to gain insight into the possible topological signature of the
chiral-spin phase by calculating the dynamic structure factor and the modular
$\mathcal{S}$ matrix.
