ct.category theory - A toy example of a tensor triangulated category?

Not quite an answer.If you want to get a better understand what is going about reconstruction theorem, maybe you could take a look at this quesHow to unify various reconstruction theoremstion I asked.

I think the first step to understand these constructions is to take some really "trivial" example, such as $A-mod$,(I assume $A$ is commutative noetherian ring,but actually in abelian level, we do not need noethrian). Then you consider bounded derived category of $A-mod$, i.e. $D^b(A-mod)$. Take them as symmetric monoidal category. Then, calculate the spectrum of this triangulated category and then do the geometric realization.

Another example you can consider is $D^b(CohP^1)$. However, from my understanding, there is nothing much you can calculate. Because for the symmetric tensor triangulated category, the spectrum gets much simpler than non-symmetric case. It is direct imitation of prime spectrum of a commutative ring.

Moreover, I am not sure whether P.Balmer's reconstruction theorem works for non-noetherian case. (In abelian level, it does)