  Electrostatics on global Anti-de-Sitter (AdS) spacetime is sharply different
from that on global Minkowski spacetime. It admits a multipolar expansion with
everywhere regular, finite energy solutions, for every multipole moment except
the monopole (arXiv:1507.04370). A similar statement holds for global AdS
magnetostatics. We show that everywhere regular, finite energy, electric plus
magnetic fields exist on AdS in three distinct classes: $(I)$ with
non-vanishing total angular momentum $J$; $(II)$ with vanishing $J$ but
non-zero angular momentum density, $T^t_\varphi$; $(III)$ with vanishing $J$
and $T^t_\varphi$. Considering backreaction, these configurations remain
everywhere smooth and finite energy, and we find, for example,
Einstein-Maxwell-AdS solitons that are globally - Type I - or locally (but not
globally) - Type II - spinning. This backreaction is considered first
perturbatively, using analytical methods and then non-perturbatively, by
constructing numerical solutions of the fully non-linear Einstein-Maxwell-AdS
system. The variation of the energy and total angular momentum with the
boundary data is explicitly exhibited for one example of a spinning soliton.
