  We prove global equivariant refinements of Miller's stable splittings of the
infinite orthogonal, unitary and symplectic groups, and more generally of the
spaces $O/O(m)$, $U/U(m)$ and $Sp/Sp(m)$. As such, our results encode
compatible equivariant stable splittings, for all compact Lie groups, of
specific equivariant refinements of these spaces.
  In the unitary and symplectic case, we also take the actions of the Galois
groups into account. To properly formulate these Galois-global statements, we
introduce a generalization of global stable homotopy theory in the presence of
an extrinsic action of an additional topological group.
