This question is related to the question: Is there a $k$-structure for Hodge modules over a $k$-variety?.

Suppose $K$ is a subfield of $\mathbb{C}$ and $M$ is a holonomic $D$-module "of geometric origin" on a smooth scheme $X/K$. By Saito's theory, the base-changed $D$-module $M_{\mathbb{C}}$ on $X\underset{\operatorname{Spec}(K)}\times \operatorname{Spec}(\mathbb{C})$ carries a natural mixed Hodge module structure.

An ill-formed question: how does the $K$-rational structure on $M_{\mathbb{C}}$ interact with the Hodge structures?

Here are some precise incarnations of that question. Saito's theory endows $M_{\mathbb{C}}$ with the weight filtration $W$ (by $D$-submodules) and a Hodge filtration (compatible with the filtration on differential operators). Are $W$ and $F$ defined over $K$? Or, a seemingly weaker question: does the action of $\operatorname{Gal}(\mathbb{C}/F)$ on (the mixed Hodge module of) de Rham cohomology of $M_{\mathbb{C}}$ preserve the induced filtrations on de Rham cohomology? (This is readily verified when $M=\mathcal{O}_X$).