TITLE: Are "large enough" finite etale covers arithmetic?
QUESTION [9 upvotes]: Let $X$ be a variety over a number field $K$. Then it is known that for any topological covering $X' \to X(\mathbb{C})$, the topological space $X'$ can be given the structure of a $\overline{K}$-variety in such a way so that the morphism $f: X' \to X$ inducing the topological map is a finite etale morphism over $\overline{K}$. However, the variety $X'$ and the morphism $f$ may not descend to $K$.
My question is as follows: does there always exist a further finite etale covering $f' : X'' \to X'$ such that the composition $X'' \to X$ may be defined over $K$?
EDIT: Just to be clear, I'd like all the covers involved to be geometrically connected to avoid trivial solutions.

REPLY [9 votes]: Adding on Will's and Sasha's answers, the condition of having a rational point, or at least a "1-truncated homotopy fixed point" for the action is necessary. For example, let $C_2$ act on the circle $S^1$ by half rotation. The covers of $S^1$ are the standard n-fold ones, and we can ask what it takes to lift the action of $C_2$ to the cover, so that it is "defined over $BC_2$". In particular, we need to lift that half-circle rotation to the n-fold cover, for which the options are $1/2n + k/n$ rounds rotation. For this to be an involution, we need that applying it twice gives the identity, i.e. that $1/n +2k/n$ is an integer. If $n$ is even, this is impossible, and so the double cover of this action on $S^1$ has no cover definable over $BC_2$. To turn this topological picture to arithmetic, take $K=\mathbb{R}$ and let complex conjugation act on $\mathbb{C}^\times$ by $z\mapsto -1/\bar{z}$ (which is a form of the multiplicative group with no rational points). The action on the unit circle is then half rotation, so the Galois story realized to the topological one up to profinite completion.
I would add that what happens topologically is that if we have a fixed point, we can use it to define a "connected" compositum of pointed covers, by taking the component of the tuple of base points lifts. This is what missing in this example esssentially, even though up to isomorphism all covers are actually "the same".