The way to study the topology of the situation was introduced by Khovanski in
"Newton polyhedra, and toroidal varieties" Funkcional. Anal. i Priložen. 11
(1977), no. 4, 56--64, 96. His result (if I have interpreted it correctly) is
that $X$ may be compactified as a hypersurface in a projective toric variety to
a smooth variety with normal crossings such that each stratum is of the same
form as $X$. As far as I can see this construction works uniformly so that we
would get the same construction over a (suitable) mixed characteristic discrete
valuation ring. Then desired isomorphism then follows from the smooth and proper
base change theorem. (I have some vague recollection that this comment is also to be
found somewhere in SGA but I am not going to do any wading looking for it...)