  We study the tensor-triangular geometry of the category of equivariant
$G$-spectra for $G$ a profinite group, $\mathsf{Sp}_G$. Our starting point is
the construction of a ``continuous'' model for this category, which we show
agrees with all other models in the literature. We describe the Balmer spectrum
of finite $G$-spectra up to the ambiguity that is present in the finite group
case; in particular, we obtain a thick subcategory theorem when $G$ is abelian.
By verifying the bijectivity hypothesis for $\mathsf{Sp}_G$, we prove a
nilpotence theorem for all profinite groups.
  Our study then moves to the realm of rational $G$-equivariant spectra. By
exploiting the continuity of our model, we construct an equivalence between the
category of rational $G$-spectra and the algebraic model of the second author
and Sugrue, which improves their result to the symmetric monoidal and
$\infty$-categorical level. Furthermore, we prove that the telescope conjecture
holds in this category. Finally, we characterize when the category of rational
$G$-spectra is stratified, resulting in a classification of the localizing
ideals in terms of conjugacy classes of subgroups.
  To facilitate these results, we develop some foundational aspects of
pro-tt-geometry. For instance, we establish and use the continuity of the
homological spectrum and introduce a notion of von Neumann regular
tt-categories, of which rational $G$-spectra is an example.
