  For $J$-hermitian operators on a Krein space $(\mathcal{K},J)$ satisfying an
adequate Fredholm property, a global Krein signature is shown to be a homotopy
invariant. It is argued that this global signature is a generalization of the
Noether index. When the Krein space has a supplementary Real structure, the
sets of $J$-hermitian Fredholm operators with Real symmetry can be retracted to
certain of the classifying spaces of Atiyah and Singer. Secondary
$\mathbb{Z}_2$-invariants are introduced to label their connected components.
Related invariants are also analyzed for $J$-unitary operators.
