  The mixedness of a quantum state is usually seen as an adversary to
topological quantization of observables. For example, exact quantization of the
charge transported in a so-called Thouless adiabatic pump is lifted at any
finite temperature in symmetry-protected topological insulators. Here, we show
that certain directly observable many-body correlators preserve the integrity
of topological invariants for mixed Gaussian quantum states in one dimension.
Our approach relies on the expectation value of the many-body
momentum-translation operator, and leads to a physical observable --- the
"ensemble geometric phase" (EGP) --- which represents a bona fide geometric
phase for mixed quantum states, in the thermodynamic limit. In cyclic
protocols, the EGP provides a topologically quantized observable which detects
encircled spectral singularities ("purity-gap" closing points) of density
matrices. While we identify the many-body nature of the EGP as a key
ingredient, we propose a conceptually simple, interferometric setup to directly
measure the latter in experiments with mesoscopic ensembles of ultracold atoms.
