Is there any theory of "étale cohomology" with algebraic coefficients?

Question

For simplicity, I will restrict attention to untwisted coefficients.

Let $k$ be a finite field of characteristic $p$, and $\ell\ne p$ prime. Can one define a cohomology theory with $\mathbb{Q}_\ell^{\mathrm{alg}}:=\overline{\mathbb{Q}}\cap\mathbb{Q}_\ell$ coefficients? More precisely, are there contravariant functors $X\mapsto H^i(X;\mathbb{Q}_\ell^{\mathrm{alg}})$ from $k$-varieties to representations of $\mathbb{Z}\subseteq\operatorname{Gal}(k)$ on graded-commutative $\mathbb{Q}_\ell^{\mathrm{alg}}$-algebras such that $H^i_{\mathrm{et}}(X;\mathbb{Q}_\ell)=H^i(X;\mathbb{Q}_\ell^{\mathrm{alg}})\otimes_{\mathbb{Q}_\ell^{\mathrm{alg}}}\mathbb{Q}_\ell$?

A few observations:

1. It is well-known that one cannot define "étale" cohomology with coefficients in an arbitrary field of characteristic zero. The obstruction I know (I think this is due to Serre) is if $E$ is a supersingular elliptic curve, then the cohomology of $E$ with coefficients in $F=\mathbb{Q}_p$ or $\mathbb{R}$ would have to form a module over $\operatorname{End}(E)\otimes F$ of $F$-dimension 2, which does not exist. However, this obstruction "and anything of this nature" are insensitive to elementary equivalence, so by a theorem of Ax and Kochen they can't exclude $\mathbb{Q}_\ell^{\mathrm{alg}}$-coefficients.
2. Basic completeness considerations eliminate any representation of the full Galois group $\hat{\mathbb{Z}}$ (say, discrete topology). This also presents a problem for other base fields, at least if you want functoriality in the base field or a Galois action.
3. This would imply a self-map of a variety acts on cohomology only with algebraic eigenvalues. I think this is fairly straightforward to prove using fancy theorems, but it's been too long a day to be certain; in any case, this is a meaty enough claim that any construction should have to be fairly substantive. (Related: my initial motivation was algebraicity of Frobenius eigenvalues.)
4. Motivic people probably know a lot more about this kind of question than I do, which is why I am asking here.

Answer 1

Let $M(k)$ be the category of mixed motives over the finite field $k$ (with $\mathbb{Q}$-coefficients). I think you are asking for a fibre functor with values in a number field $Q$, possibly of infinite degree, that admits a homomorphism $Q\to \mathbb{Q}_{\ell}$. Conjecturally every motive over $k$ is a direct sum of pure motives, so we replace $M(k)$ with the category of pure motives, and we assume the Tate and standard conjectures. Let $M'$ be the tannakian subcategory of $M(k)$ generated by a finite collection of varieties. Then there exists a fibre functor on $M'$ with values in a number field $Q$ of finite degree admitting a homomorphism to $\mathbb{Q}_{\ell}$. The point is that the only obstruction to the existence of such a fibre functor comes from the endomorphism algebras of the motives, which we understand (cf. the statement (*) on p.441 of Milne's 1994 article Motives over a Finite Field). Whether this can be extended to the whole category, I don't know.