Classical (i.e., discrete) logic is well positioned to study imaginaries in part because the $T^{eq}$ construction allows us to treat imaginary sorts as we would treat any other sort. With hyperimaginaries, on the other hand, classical logic has no such luck. The ability to apply a $T^{eq}$-like construction to the study of hyperimaginaries motivated, in part, the study of positive model theory, a thematic precursor to the modern approach to continuous logic.

Despite that history, I've been unable to convince myself that continuous logic's $T^{eq}$ construction, when applied to a classical theory, includes sorts for the classical hyperimaginaries (even finitary ones) in the case of an uncountable theory. (The $T^{eq}$ construction I'm using is the one that adds canonical parameters for continuous logic formulas, such as in Chapter 11 of Model Theory for Metric Structures.)

Folklore as I've heard it says that the construction does include all the classical finitary hyperimaginaries, but I've been unable to find a citation for that fact, and haven't yet produced a clear proof or counterexample. Can anyone point me in the right direction?