Let $H$ be a mixed Hodge structure or, more generally, a mixed Hodge structure over a subfield $k$ of $\mathbb{C}$, by which I mean a $k$-vector space with two filtrations (Hodge and weight), a $\mathbb{Q}$-vector space with a filtration (weight) and an isomorphism between the complexifications such that weight filtrations correspond to each other and these data induce a mixed Hodge structure on the $\mathbb{Q}$-vector space.

The main example is given by the cohomology of an algebraic variety over $k$. This and all their sub Hodge structures certainly have geometric origin. But I guess there should be more, for example I want the cohomology of a variety with values in the local system $R^n f_\ast \mathbb{Q}$ for some morphism of varieties $f \colon X \to Y$ to be of geometric origin as well.

What is the general definition? Is there common agreement about it?