This is a reference request. Are there any results of the following kind?
Assume $\mathrm{CH}$ and let $\mathcal{U}$ be a Ramsey ultrafilter. Let $c$ be a Cohen real. Then in $V[c]$, can $\mathcal{U}\cup\{c\}$ be extended to a Ramsey ultrafilter again? Instead of a Cohen real, any results concerning other splitting reals would work too. This seems basic enough, but I couldn't find anything yet.
Note that I am looking for positive answers. As for negative answers, there is for example Kunen's result that rapid p-points can't be extended to p-points after adding any number of random reals.
