By introducing a suitable range-separation of the Coulomb coupling in analogy to Ambrosetti et al (2014 J. Chem. Phys. 140 18A508), here we extend the many-body dispersion approach to include beyond-dipole van der Waals (vdW) interactions at a full many-body level, in combination with semi-local density functional theory. A reciprocal-space implementation is further introduced in order to efficiently treat periodic systems. Consistent reliability is found from molecular dimers to large supramolecular complexes and two-dimensional systems. The large weight of both many-body effects and multipolar terms illustrates how a correct description of vdW forces in large-scale systems requires full account of both contributions, beyond standard pairwise dipolar approaches.