In the study of stability we usually work with $\mathbb{M}^{eq}$ which allows us to treat each equivalence class of a definable equivalence relations as an element.
In his book on simple theories, Wagner states that the analogues process of adding hyperimaginaries (essentially adding elements for each equivalence class of type definable equivalence relations) elements to form $\mathbb{M}^{heq}$ fails. He provides an example of a theory (the unit circle with $R_{n}(x,y)$ if and only if the distance between $x,y$ is less than $1/n$) for which this process fails.
However Wagner himself notes that the example is not simple. And other texts on simple theories seem to just consider $\mathbb{M}^{heq}$ as a given. My questions are:
1) Is there any reason that $\mathbb{M}^{heq}$ can be defined for simple theories without running in to contradictions?
2) Is there a way of defining $\mathbb{M}^{heq}$ for arbitrary theories?
3) Or is Wagner saying while we can form a $\mathbb{M}^{heq}$ with the elements, we cannot form a corresponding $T^{heq}$?
Edit 1: In response to Alex's comment I'm going to post the details of the example (3.1.6 of Wagner's text). I'm very slightly paraphrasing what Wagner says in his text. Hopefully I get this down correctly:
Let $\mathcal{M}$ be the unit circle and assume that $R_n$ is as above. Let $E(x,y)$ be the type definable equivalence relation that holds on $x,y$ if and only if $x,y$ are infinitely close together (i.e. $\bigwedge R_n(x,y)$). Let $\mathcal{N}$ be an elementary extension of $\mathcal{M}$. Now given any element $n$ of $N$, the equivalence class is given by the standard part $std(n)\in \mathcal{M}$. So $E$ has precisely continuum many classes.
Now if we could add a predicate for $\mathcal{M}/E$ and a definable map $\pi_E:x\rightarrow x_E$, then compactness would imply that there are only finitely many $E$-classes.
3) is correct. There's no problem with forming $M^{\text{heq}}$ for an arbitrary model $M$ of an arbitrary theory $T$: just add a new sort for every type-definable equivalence relation $E$, populate it with the equivalence classes for $E$, and add the quotient map.
The point of Wagner's example is that compactness fails for these hyperimaginary sorts, which implies that in general there is no first-order theory $T^{\text{heq}}$ whose models are exactly those of the form $M^{\text{heq}}$ where $M\models T$.
It's still quite useful, especially in simple theories, to work with these sorts. But one has to be careful to remember not to use compactness!