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][1].)

**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?

  [1]: https://www.researchgate.net/profile/Alexander_Berenstein/publication/255669163_Model_Theory_for_Metric_Structures/links/5548de4d0cf25a87816aa938.pdf