  We shed light on the interplay between structure and many-body effects
relevant for itinerant ferromagnetism in LaAlO$_3$/SrTiO$_3$ heterostructures.
The realistic correlated electronic structure is studied by means of the
(spin-polarized) charge self-consistent combination of density functional
theory (DFT) with dynamical mean-field theory (DMFT) beyond the realm of static
correlation effects. Though many-body behavior is also active in the
defect-free interface, a ferromagnetic instability occurs only with oxygen
vacancies. A minimal Ti two-orbital $e_g$-$t_{2g}$ description for the
correlated subspace is derived. Magnetic order affected by quantum fluctuations
builds up from effective double exchange between nearly-localized $e_g$ and
mobile $xy$ electrons.
