  We examine how the ability of a system to redundantly proliferate relevant
information about its pointer states is affected when it is coupled to multiple
baths. To this end, we consider a system in contact with two baths: one --
termed the {\it accessible} environment -- which, on its own, induces a pure
dephasing mechanism on the state of the system and satisfies the conditions for
classical objectivity to be established. The second environment, which we dub
as {\it inaccessible}, affects the system in two physically relevant ways.
Firstly, we consider an interaction that commutes with the Hamiltonian
describing the interaction between system and accessible bath. It thus also
gives rise to dephasing of the system, albeit on different time scales.
Secondly, we consider a thermalising interaction, which does not commute with
the system-accessible environment Hamiltonian. While the former still allows
the system to redundantly encode its state into the accessible environment, the
latter degrades the correlations, eventually destroying them in the long-time
limit, and thus leads to a loss of the conditions necessary for classical
objectivity to be established. This sheds light on the role that commutativity
between the various system-bath interaction terms plays when establishing the
conditions for classical objectivity to be supported.
