Consistency of a strange (choice-wise) set of reals

Consider a set $X\subseteq \mathbb{R}$ such that

1. $X$ is not separable wrt its subspace topology
2. For all $r\in\mathbb{R}$ there exists a sequence $(x_n)_{n\in\omega} \subset X$ converging to $r$

In a model containing such a set $\text{AC}_\omega(X)$ (choice for countable families of non-empty subsets of $X$) would of course fail, but not that critically.

For example, the unique way I've seen to prove the consistency of the existence of a non-separable set of reals is the one that shows the consistency of an infinite, Dedekind-finite set of reals, but our set, though being non-separable, is well-behaved enough to witness density in a sequencial manner.

My questions are:

• Is its existence consistent relative to $\text{ZF}$? Has it been proved somehwere?
• In case the answer above is "no", does this remind you similar results (besides the most known ones that can be found in Jech'Axiom of Choice)?

Thanks!