In the context of quantum theory, suppose we have two models $M_1$ and $M_2$ formulated on the same Hilbert space. Suppose that the operator $A$ is an observable in both models, with the "same" physical interpretation in both models. Suppose also that the operator $B$ is an observable in both models, with the "same" physical interpretation in both models.
Question: Can $A+B$ be an observable in both models with a physical interpretation in $M_1$ that differs from its interpretation in $M_2$?
My gut says yes, $A+B$ can have different interpretations in the two models, but I haven't come up with a satisfying example.
The question assumes that we're using a systematic approach (not ad-hoc) for associating physical observables to self-adjoint operators, like what is used in quantum field theory. I considered adding the quantum-field-theory tag for that reason, but I decided against that because there could be other systematic approaches, too.
This question was inspired by the question What is the physical meaning of the sum of two non-commuting observables?, but this new question is posed differently to solicit a specific kind of answer, namely answers that compare two specific models, rather than merely addressing generalities about measurement.
In the phrase "same" physical interpretation, I put "same" in scare-quotes because one could question whether or not that's a meaningful concept, but I'm hoping to avoid that issue here.
I'm not asking whether or not $A+B$ is necessarily an observable. That might also be an interesting question, but here I'm only considering models in which $A+B$ is taken to be an observable.
I'm also not asking whether or not we have any hope of measuring $A+B$ in practice. I suspect there are cases where measuring $A+B$ would hopelessly complicated even if measuring $A$ and $B$ individually is simple, and that thought motivated me to post a related question on Math SE (https://math.stackexchange.com/q/3334439). For the present question, measurability in principle is sufficient. I admit that the distinction between practice and principle might itself be a non-trivial issue (cf chapter 7 in Omnes , The Interpretation of Quantum Mechanics, Princeton University Press: "Some observables cannot be measured, even as a matter of principle"), but I'm hoping the present question can be answered without getting into that.