Let $R$ be a DVR of char $0$ and $S=Spec(R)$. Let $X\longrightarrow S$ be a proper flat morphism. Assume $X$ is integral. De Jong's Theorem 8.2 in his paper Smoothness, semi-stability and alterations implies that $X$ can be altered into a scheme $Y$ which is semi-stable over a DVR $R_1$ where $R_1$ is finite over $R$. Does the above statement still hold if we require $K(Y)/K(X)$ is a finite Galois extension?
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$\begingroup$ Very roughly, the field $K(Y)$ only depends on the generic fiber of $Y$, and the extension $K(Y)/K(X)$ will be Galois if and only if $\text{Frac}(R_1)//\text{Frac}(R)$ is Galois. Now it suffices to note that there exists a semistable model over any finite extension of $R_1$, for which we can take the Galois closure of the fraction field of $R_1$. $\endgroup$– JefCommented Apr 22, 2020 at 14:15
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This is a special case of Lemma 3.8 of the paper "Sur l'indépendance de l en cohomologie l-adique sur les corps locaux". (Sanity check: in the proof of 4.11 the author uses that $Y$ obtained from Lemma 3.8 is strictly semi-stable over the new dvr. Also, since you assumed the fraction field of your dvr has char 0 there is no issue with inseparability of the extensions of fields we obtain.)
Many other authors have proved different and interesting versions of the original result by de Jong. In particular, I mention Gabber's results (see for example "Traveaux the Gabber sur l'uniformisation...") and very strong techinical results by Temkin. I'm sure you can deduce what you want also from those papers, but as far as I can tell it would be a less direct reference. Cheers!
