It is straightforward to construct proper uncountable subgroups of $\mathbb{R}$. One can construst a basis for $\mathbb{R}$ over $\mathbb{Q}$, and then there are many possibilities (just consider the group generated by the basis or the vector subspace generated by some proper uncountable set of the basis).

However, the first step (constructing the basis) requires the axiom of choice.

So does anyone know of any proper uncountable subgroup of $\mathbb{R}$ that does not require choice to construct?

or is this not possible.

Meaning are there models not involving choice where every uncountable subgroup of $\mathbb{R}$ is equal to $\mathbb{R}$.