  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.
