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The Galois group $Gal(\bar{\mathbb Q}/\mathbb Q)$ is a profinte group, thus compact. For each $p$, take a representative of $Frob_p$ in that group. Thre ultrafilter produces an element $Frob_u$, whose trace in the $l$-adic Galois representation at a smooth prime $l$ (a continuous representation) is some number $a_u$, which we can express as $2 \sqrt{u} \cdot \cos (\theta_u)$, with $u$ and $\theta_u$ both $l$-adic numbers. However these are $l$-adic versions, not real.