Stratifications of tangent cones in real closed (valued) fields

We introduce tangent cones of subsets of cartesian powers of a real closed field, generalising the notion of the classical tangent cones of subsets of Euclidean space. We then study the impact of non-archimedean stratifications (t-stratifications) on these tangent cones. Our main result is that a t-stratification induces stratifications of the same nature on the tangent cones of a definable set. As a consequence, we show that the archimedean counterpart of a t-stratification induces Whitney stratifications on the tangent cones of a semi-algebraic set. The latter statement is achieved by working with the natural valuative structure of non-standard models of the real field.