I'll express the relationship I have in mind in something like model theory, but there might be analogous relationships in abstract algebras or category theory. I'm looking for a name for the relationship between two theories that both have the same observable consequences on observable objects, even though they each use different sets of theoretical objects with different theoretical properties.
Here is the relationship in abstract: Suppose you have a set $R$ of "real" objects, and two sets $U_1 \supset R$ and $U_2 \supset R$ of real and theoretical objects. There are two theories (sets of propositions closed under deduction), $T_1$ and $T_2$ that refer only to objects of $U_1$ and $U_2$, respectively. Let's use the notation that $T \backslash U$ is the set of propositions in $T$ that are only about objects in $U$. That is, it is the restriction of $T$ to a smaller set of objects.
I'm looking for a word for the relationship between $T_1$ and $T_2$ when $T_1 \backslash R = T_2 \backslash R$.