The answers to your question in the case $n=1$ are well-known. In higher dimensions, the answers are less complete, but something is known.

For example, in the real case when $n=1$, there is only one smooth, connected compact $1$-manifold, the circle, and, for each natural number $k\ge1$, there is a rational structure $\mathcal{R}_k$, which is the rational structure induced on the $k$-fold connected cover of $\mathbb{RP}^1\simeq S^1$ (which is smoothly diffeomorphic to $S^1$, of course). Two rational structures $\mathcal{R}_i$ and $\mathcal{R}_j$ on the circle are isomorphic if and only if $i=j$, and every rational structure on the circle is isomorphic to some $\mathcal{R}_i$.

Meanwhile, in the complex case when $n=1$, specifying a rational structure on a compact Riemann surface of genus $g$ (i.e., a topological surface of genus $g$ with a fixed underlying holomorphic structure) is easily seen to be equivalent to specifying a projective structure on the surface. For $g>1$, it is known that the moduli of projective structures on a given compact Riemann surface is equivalent to the moduli of quadratic holomorphic differentials, a complex vector space of dimension $3g{-}3$. For example, see R. C. Gunning's On uniformization of complex manifolds: The role of connections. In particular, these provide counterexamples sought by the OP.

In higher dimensions, for example, when $n=2$, the question of when a given compact complex surface has a rational structure is well-understood since we have a classification of compact complex surfaces, by the work of Kodaira. Not all such surfaces have a rational structure; for example, a K3 surface does not.

The answers to your question in the case $n=1$ are well-known. In higher dimensions, the answers are less complete, but something is known.

For example, in the real case when $n=1$, there is only one smooth, connected compact $1$-manifold, the circle, and, for each natural number $k\ge1$, there is a rational structure $\mathcal{R}_k$, which is the rational structure induced on the $k$-fold connected cover of $\mathbb{RP}^1\simeq S^1$ (which is smoothly diffeomorphic to $S^1$, of course). Two rational structures $\mathcal{R}_i$ and $\mathcal{R}_j$ on the circle are isomorphic if and only if $i=j$, and every rational structure on the circle is isomorphic to some $\mathcal{R}_i$.

Meanwhile, in the complex case when $n=1$, specifying a rational structure on a compact Riemann surface of genus $g$ (i.e., a topological surface of genus $g$ with a fixed underlying holomorphic structure) is easily seen to be equivalent to specifying a projective structure on the surface. For $g>1$, it is known that the moduli of projective structures on a given compact Riemann surface is equivalent to the moduli of quadratic holomorphic differentials, a complex vector space of dimension $3g{-}3$. For example, see R. C. Gunning's On uniformization of complex manifolds: The role of connections. In particular, these provide counterexamples sought by the OP.

In higher dimensions the situation is more complicated because the pseudogroup of rational maps with a rational inverse, i.e., the so-called birational pseudogroup, in either $\mathbb{R}^n$ or $\mathbb{C}^n$ is not well-understood when $n>1$.

However, when $n=2$, the question of when a given compact complex surface has a rational structure is well-understood since we have a classification of compact complex surfaces, by the work of Kodaira. Not all such surfaces have a rational structure; for example, a K3 surface does not.