There appears to be a characterization of equivalence in stationary logic by means of Ehrenfeucht–Fraı̈ssé-like games. For instance, this appears to have been studied by Caicedo, Kaufmann, Makowsky, Seese, and Weese. I found these in citations, but the cited versions of their results are either in theses or unpublished manuscripts.
Are there actually published versions of these results? (As a bonus, is there any other semantic characterizations of equivalence in L(aa)?)