Why $\overline{\mathbb{Q}_\ell}$?

Question

In étale cohomology usually the only well-behaving coefficient sheaves are the finite ones. But one needs a characteristic zero coefficient sheaf for the fixed-point formula to work, thus the $\mathbb{Z}_\ell$. One wants a field coefficient so that the cohomologies are free and the trace can be defined, thus $\mathbb{Q}_\ell$. But in Kiehl-Weissauer usually $\overline{\mathbb{Q}_\ell}$ is used; I've heard that it is for cohomological comparison. Can you elaborate what comparison theorems are present here? Thanks!