Interesting examples of $Q$-fiber bundles

Question

Let $X$ be a topological space. What are some interesting examples of $\mathbb{Q}$-fiber bundles (fiber bundles with fiber some $\mathbb{Q}$-vector space, though not necesarily vector bundles) on $X$? Are most of these examples just given by studying $\mathbb{R}$-fiber bundles on $X$ by density considerations or extension of scalars in the vector bundle case? The examples I'm thinking of are hopefully unexpected, and connected to other problems, such as the $\mathbb{Q}$-bundle structure in the sequence of topological groups $\mathbb{Q} \hookrightarrow{} \mathbb{A} \to \mathbb{A}/\mathbb{Q} \text{ }$ described in this paper of Burgos and Verjovsky, where $\mathbb{A}$ is the Adele group of the rationals, which is used to describe the adelic solenoid $S_{\mathbb{Q}}^{1} \cong \mathbb{A}/\mathbb{Q}$.