  We examine the emergence of objectivity for quantum many-body systems in a
setting without an environment to decohere the system's state, but where
observers can only access small fragments of the whole system. We extend the
result of Reidel (2017) to the case where the system is in a mixed state,
measurements are performed through POVMs, and imprints of the outcomes are
imperfect. We introduce a new condition on states and measurements to recover
full classicality for any number of observers. We further show that evolutions
of quantum many-body systems can be expected to yield states that satisfy this
condition whenever the corresponding measurement outcomes are redundant.
