  The presence of spin-orbit coupling or non-collinear magnetic spin states can
have dramatic effects on the ground-state and spectral properties of materials,
in particular on the band structure. Here, we develop non-collinear
Koopmans-compliant functionals based on Wannier functions and
density-functional perturbation theory, targeting accurate spectral properties
in the quasiparticle approximation. Our non-collinear Koopmans-compliant theory
involves functionals of four-component orbitals densities, that can be obtained
from the charge and spin-vector densities of Wannier functions. We validate our
approach on four emblematic non-magnetic and magnetic semiconductors where the
effect of spin-orbit coupling goes from small to very large: the III-IV
semiconductor GaAs, the transition-metal dichalcogenide WSe$_2$, the cubic
perovskite CsPbBr$_3$, and the ferromagnetic semiconductor CrI$_3$. The
predicted band gaps are comparable in accuracy to state-of-the-art many-body
perturbation theory and include spin-dependent interactions and screening
effects that are missing in standard diagrammatic approaches based on the
random phase approximation. While the inclusion of orbital- and spin-dependent
interactions in many-body perturbation theory requires self-screening or vertex
corrections, they emerge naturally in the Koopmans-functionals framework.
