  We prove a reconstruction theorem valid for arbitrary theories in continuous
(or classical) logic in a countable language, that is to say that we provide a
complete bi-interpretation invariant for such theories, taking the form of an
open Polish topological groupoid. More explicitly, for every such theory $T$ we
construct a groupoid $\mathbf{G}^*(T)$ that only depends on the
bi-interpretation class of $T$, and conversely, we reconstruct from
$\mathbf{G}^*(T)$ a theory that is bi-interpretable with $T$. The basis of
$\mathbf{G}^*(T)$ (namely, the set of objects, when viewed as a category) is
always homeomorphic to the Lelek fan. We break the construction of the
invariant into two steps. In the second step we construct a groupoid from any
\emph{reconstruction sort}, while in the first step such a sort is constructed.
This allows us to place our result in a common framework with previously
established ones, which only differ by their different choice of a
reconstruction sort.
