  We study an exactly solvable model with bond-directional quadrupolar and
octupolar interactions between spin-orbital entangled $j_{\mathrm{eff}} =
\frac{3}{2}$ moments on the honeycomb lattice. We show that this model features
a multipolar spin liquid phase with gapless fermionic excitations. In the
presence of perturbations that break time-reversal and rotation symmetries, we
find Abelian and non-Abelian topological phases in which the Chern number
evaluates to $0$, $\pm 1$, and $\pm 2$. We also investigate quantum phase
transitions out of the multipolar spin liquid using a parton mean-field
approach and orbital wave theory. In the regime of strong
integrability-breaking interactions, the multipolar spin liquid gives way to
ferroquadrupolar-vortex and antiferro-octupolar ordered phases that harbor a
hidden spin-$\frac{1}{2}$ Kitaev spin liquid. Our work unveils mechanisms for
unusual multipolar orders and quantum spin liquids in Mott insulators with
strong spin-orbit coupling.
