Partial derivatives "split" over $\mathbb R$

Kaplansky's book Fields and Rings, page 30, implicitly contains the following question.

For which fields $K$ does the following statement hold true for every polynomial $f \in K[X]$: If $g$ splits over $K$ then its derivative $g'$ splits over $K$.

This is, for instance, true for real closed fields, e.g., for $\mathbb R$ and the paper A weak version of Rolle's Theorem by Thomas C. Craven in Proc. AMS 125:11, p. 3147–3153, contains further treatment of the problem.

I am interested in the multivariate analogue of this problem over $\mathbb R$ and I could not find anything about it. Let me make the question precise:

Question: Let $g \in \mathbb R[X_1,\ldots,X_n]$ be a polynomial that factors over $\mathbb R$ into polynomials that are irreducible over $\mathbb C$. Do all partial derivatives $\frac{\partial g}{\partial X_i}$ factor over $\mathbb R$ into polynomials irreducible over $\mathbb C$.

This arises naturally in real birational geometry. The following can be made precise and leads to the above question: Are certain hypersurfaces that are normal crossing over $\mathbb R$ also normal crossing over $\mathbb C$?

Thanks in advance for your help!