In previous work, we have established a generalisation of the so-called Ahlbrandt-Ziegler principle, associating to a complete first-order theory $T$ (in a countable, Boolean-valued language) a Polish groupoid $G(T)$, which acts as a complete bi-interpretation invariant for $T$.

The construction of $G(T)$ does not extend directly to continuous first-order logic (real-valued, with semantics on metric structures), essentially since one cannot always choose witnesses for formulas in a continuous fashion. A less-than-satisfactory solution was proposed, namely to assume that the exists a sufficiently rich sort of potential parameters for which witnesses can be chosen, covering some, but not all, cases.

We propose a better solution, that covers all cases, at the cost of replacing spaces with cones over them. In particular, totally disconnected spaces of types are replaced with fans.