$\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\SH{SH}\DeclareMathOperator\Mod{Mod}\newcommand{\fin}{\mathrm{fin}}\newcommand{\cc}{\mathrm{c}}$I'm mainly interested in the dynamics of the Balmer spectrum construction at the moment. I know that famously the Balmer spec of a derived category of a large number of schemes recovers the original space, but it is known that the construction only results in a quasi-compact locally ringed space, which might not be a scheme for an arbitrary tensor triangulated category. Balmer gives the following result in
Balmer, Paul, Spectra, spectra, spectra – tensor triangular spectra versus Zariski spectra of endomorphism rings, Algebr. Geom. Topol. 10, No. 3, 1521-1563 (2010). ZBL1204.18005.
Proposition 9.7. The locally ringed space $\Spec(\SH^{\fin})$ is not a scheme. Nor is any of the local ones $\Spec(\SH^{\fin}_{(p)})$, for any prime p.
Where $\SH^{\fin}$ is the stable homotopy category of finite spectra.
My question is: Let $D=D(\Mod(\Spec(\SH^{\fin}))$ be the derived category of sheaves of modules over $\Spec(\SH^{\fin})$, this category is a tensor triangulated category. Is it true that $\Spec(D^{\cc})\cong \Spec(\SH^{\fin})$?
Here $D^{\cc}$ is the subcategory of compact objects which I assume is also a tensor triangulated category.
I suspect that the answer is either no or that this is too hard to calculate in particular. The spirit of my question, however, is more in the lines of the following:
Let $T$ be a compactly generated tensor triangulated category and $K=T^{\cc}$, when is $\Spec(D(\Spec(K)^{\cc})\cong \Spec(K)$? Obviously if $D(\Spec(K))^{\cc}\cong K$ as tensor triangulated categories this works, but are there other situations?