  Layered material $\alpha$-RuCl$_3$ has caught wide attention due to its
possible realization of Kitaev's spin liquid and its electronic structure that
involves the interplay of electron-electron correlations and spin-orbit
effects. Several DFT$+U$ studies have suggested that both electron-electron
correlations and spin-orbit effects are crucial for accurately describing the
band gap. This work studies the importance of these two effects using
fixed-node and fixed-phase diffusion Monte Carlo calculations both in
spin-averaged and explicit spin-orbit formalisms. In the latter, the
Slater-Jastrow trial function is constructed from two-component spin-orbitals
using our recent quantum Monte Carlo (QMC) developments and thoroughly tested
effective core potentials. Our results show that the gap in the ideal crystal
is already accurately described by the spin-averaged case, with the dominant
role being played by the magnetic ground state with significant exchange and
electron correlation effects. We find qualitative agreement between hybrid DFT,
DFT+$U$, and QMC. In addition, QMC results agree very well with available
experiments, and we identify the values of exact Fock exchange mixing that
provide comparable gaps. Explicit spin-orbit QMC calculations reveal that the
effect of spin-orbit coupling on the gap is minor, of the order of 0.2 eV,
which corresponds to the strength of the spin-orbit of the Ru atom.
