Let $T$ be a theory and let $I$ be an imaginary sort of $T$. We'll let $T\_I$ denote the induced theory on $I$, by which I mean specifically the theory of all $\varnothing$-definable relations on $I$.


Suppose that


1. for all models $\mathfrak{I} \models T\_I$ there is an $\mathfrak{M} \models T$ such that $\mathfrak{I} \cong I(\mathfrak{M})$ and furthermore $\mathfrak{M}$ is unique in the sense that for any two $\mathfrak{M}\_0$ and $\mathfrak{M}\_1$ with isomorphisms $f\_i : I(\mathfrak{M}\_i) \cong \mathfrak{I}$ there is an isomorphism $g:\mathfrak{M}\_0 \cong \mathfrak{M}\_1$ such that $$f\_1^{-1} \circ I(g) \circ f\_0 = \mathrm{id}\_\mathfrak{I},$$ where $I(g): I(\mathfrak{M}\_0)\cong I(\mathfrak{M}\_1)$ is the induced isomorphism and $\mathrm{id}\_{\mathfrak{I}}$ is the identity on $\mathfrak{I}$, and
2. the home sort of $T$ is 'extensional' over $I$ in the sense that for any $a,b \in \mathfrak{M} \models T$, we have that $a=b$ if and only if $a\equiv\_{I(\mathfrak{M})} b$.


Does it follow that the interpretation of $T\_I$ in $T$ extends to a bi-interpretation between $T$ and $T\_I$?


The conclusion does not follow from either 1 or 2 alone:


* Let $T$ be the theory of an equivalence relation with infinitely many equivalence classes of size $2$, and let $I$ be the imaginary that results from modding by the equivalence relation. $T$ and $I$ satisfy 1 but not 2 and fail the conclusion.
* Let $T$ be the theory of the collection of subsets of $\omega$ with $\subseteq$. The set of singletons is definable, so we can define an equivalence relation where $xEy$ if either $x$ and $y$ are both singletons and $x=y$ or $x$ and $y$ are both not singletons. If we let $I$ be the imaginary resulting from modding by this equivalence relation then $T$ and $I$ satisfy 2 but not 1 and fail the conclusion.



A Beth definability style result for imaginaries?
Your question lies in the realm of relative categoricity (over a predicate), which is related to the theory of stability over a predicate. Here the predicate in question is the imaginary sort $I$. 


You can read about this in Section 12.5 of Hodges's (longer) *Model Theory*. Using Hodges's terminology, your condition 1 is a bit stronger than saying that $T$ is *relatively categorical* over $I$, and conditions 1 and 2 together imply that $T$ is *rigidly relatively categorical* over $I$. 


The answer to your question is yes, at least when the language is countable. This is a consequence of Gaifman's coordinatization theorem (Theorem 12.5.8 on p. 645 of Hodges), augmented by Exercises 11 and 13 on p. 649.


