Are there examples of independence results over subsystems of true second order arithmetic that cannot be established using omega-models? To rule out trivial examples, let us assume that the base theory extends true first order arithemtic. A non example of such a statement would be Ramsey theorem for pairs since there is a computable coloring of pairs of integers into two colors without a computable (even from the halting problem, Jockusch) infinite homogeneous set.

An example of a use of non-omega models appears here - Also, see the introduction and question 6.1 in this paper.

I am not an expert in reverse mathematics so please feel free to offer any interesting known facts known to you here.