Composition Ax-Kochen/Ershov principles and tame fields of mixed characteristic

We study in which settings we have a composition AKE principle, i.e. when the theory of the coarsening $(K,w)$ and the theory of the induced valuation $(Kw,\bar{v})$ determine the theory of the composition $(K,v)$. We show that this is the case when $(K,w)$ is tame of equal characteristic, and provide counterexamples in mixed characteristic. We further show that, for a tame field of mixed characteristic, the theory of the valued field cannot, in general, be determined solely by the theories of its underlying field, its residue field, and its value group.