If you think yielding identical observations does not make two theories equivalent, then what is your criterion for equivalence of theories?

I could accept that two theories are equivalent if they yield identical observations to every possible observer, everywhere, or better yet, if they yield identical output for any given input if implemented as programs. If you write a program which simulates the laws of physics, and then you write another program which simulates "Odin" calling a function that simulates the laws of physics and doing nothing else, then I would accept that they represent equivalent theories, if they really do always result in the exact same (or isomorphic) output under every circumstance. (Though an Odin that impotent or constrained would be more of a weird programming mistake than a god.) But if the two programs don't systematically produce equivalent output for equivalent input, then they are not equivalent programs, even if none of the agents being simulated can tell the difference.