Alternatively: Reduce first to the case that the strictly henselian ring is a domain (say, by noetherian approximation to have only finitely many generic points, and then induction on the number of components). Then pass up finite covers to reduce to the case that the fraction field is algebraically closed. Then compare to the generic fibre.

Ah, this seems to be another situation where analytic adic spaces are, a priori, easier than schemes, but it can be made to work for schemes as well. The trick is to use v-descent to reduce to the case where the strictly henselian ring is in fact a valuation ring with algebraically closed fraction field. In that case, I think one can argue by comparing to the generic fibre.

@SteffenJaeschke The limit should be taken $p$-adically, not in the usual archimedean metric.

Perfect question! This is a problem that's very much on our minds, but where we feel that we do not yet have the correct approach. (In p-adic Hodge theory, the Fargues-Fontaine curve plays a central role, and similarly in complex Hodge theory it seems that the twistor-$\mathbb P^1$ plays an important role (work of Simpson, Mochizuki, ...), and ideally we'd like to develop Hodge theory in a way that makes the twistor-$\mathbb P^1$ appear organically. But we don't yet see how.)

Related in spirit, but I think a bit different, is the Ogus conjecture, discussed at mathoverflow.net/questions/389391/… (see in particular the reference to Yves Andre's "Une Introduction aux Motifs"). Ogus' conjecture is the only one I'm aware of in the neighborhood of what you are proposing.

Well, diamonds have a characteristic $p$ sheaf -- they are defined on perfectoid spaces in characteristic $p$, after all -- but no structure sheaf that, under the present analogy, should live over $\mathbb Z_p\otimes\mathbb R$. (I.e., for a diamond with two maps to $\mathrm{Spd}(\mathbb Z_p)$, there's no structure sheaf corresponding to that.)

The correct characterization is that they are the topological spaces that can be written as filtered colimits of compact Hausdorff spaces along injective transition maps. It is pretty clear that all CGWH spaces are of this form (being CG, it is the filtered colimit of the images of maps from CH spaces; and those images are themselves CH by WH); the other direction is e.g. Proposition A.14 in Schwede's "Global homotopy theory".

I don't think this is true. If $E$ is say an elliptic curve over $\mathbb Z_p$ with ordinary reduction and $G=E[p^n]$, then this extension splits only if $E$ is "close to" the canonical lift of its special fibre. In fact, usually the extension doesn't even split over the generic fibre $\mathbb Q_p$, as the corresponding Galois representation is not semisimple. In fact, if it splits over $\mathbb Q_p$, it does so over $\mathbb Z_p$ (using the valuative criterion of properness to extend the splitting).

Dear Ashutosh RC, I think it makes more sense to continue the discussion via e-mail; just contact me.

Thanks for the reference! However, for what I'm asking, I think this still goes the standard way. See in particular the reference to [17] for Lemma 3.3, and also Lemma 3.4 and its proof, which use the Cousin integral.

About Haboush: I think this is the key theorem to justify this correspondence to closed $\hat{G}$-orbits. (It guarantees that there are enough $\hat{G}$-invariant functions to distinguish between any two distinct closed $\hat{G}$-orbits.)