A real/complex rational atlas on a smooth closed manifold $M$ is an atlas with charts homeomorphic to Euclidean open sets in $\Bbb{R}^n$/$\Bbb{C}^n$ covering $M$ and real/complex rational transition maps. A real/complex rational structure is a maximal collection of compatible mentioned real/complex atlases. Two rational structures are called isomorphic if there is a topological automorphism $\sigma$ of $M$ such that each coordinate representation of $\sigma$ under a rational chart pair (from each structure) is rational. The existence of a rational structure does not imply being algebraic because every complex torus admits one (the transition maps are translations, thus polynomial), so I want to ask about how rigid these structures are.
Q$1$: Are there topological invariants classifying whether a smooth manifold admits a real/complex rational structure? Or on the contrary every smooth manifold admits one?
Q$2$: Obviously a real/complex rational structure belongs to a smooth/holomorphic structure. Consider the converse problem: what is the moduli space of real/complex rational structures compatible with the smooth/holomorphic structure on $M$? Specifically, is it true that the number of complex rational structures belonging to the holomorphic structure of $M$ is at most $1$? Counterexamples are welcome.
Q$3$: A related question exists on this website, and the first answer gives a sheaf-sense definition of rational structures. However its conclusion -- the algebraic structure of $\Bbb{C}^n$ can be pullbacked onto $M$ -- seems to contradict my complex torus example. I hope someone can help.
