Regular ω-languages with an informative right congruence

A regular language is almost fully characterized by its right congruence relation. The same does not hold for regular ω-languages. The right congruence of a regular ω-language may not be informative enough; many regular ω-languages have a trivial right congruence, and in general it is not always possible to define an ω-automaton recognizing a given language that is isomorphic to its right congruence.