The expression "Observationally indistinguishable" is sometimes used (although rarely) in the context of "Gauge symmetry" and its breakdown, Please see for example:
Ward Struyve article. This concept is related to the question here: "Gauge symmetry is not a symmetry" and used to advocate the use of gauge invariant quantities such as holonomies in the description of gauge theories.
According to this principle orbits of the gauge potentials under gauge transformations must be considered as "Observationally indistinguishable", and only gauge invariant quantities can be considered as "Observationally distinguishable".
For example, the standard heuristic explanation of the Higgs mechanism involves gauge variant quantities. Now, if gauge symmetry is not a symmetry, then what is the meaning of gauge symmetry breakdown. An explanation in Struyve's article is given through an example of the Higgs mechanism within the framework of classical field theory which involves only gauge invariant quantities.