The Tannakian reconstruction for schemes and more generally, geometric stacks from QCoh(X) with its tensor structure (due to Jacob Lurie) is explained in my answer to Tannakian Formalism . One can (and one has) try to extend this to the derived setting, where one ought to get a statement much stronger than Balmer's reconstruction theorem. First of all one needs to replace triangulated categories, which are too coarse to work with effectively, with symmetric monoidal $(\infty,1)$-categories (or symm. monoidal dg categories say in characteristic zero).

There's an obvious functor from such objects C to derived stacks Spec C given by Tannakian reconstruction -- it's defined just as in the abelian case (see my answer above). Namely we build Spec C's functor of points, as a functor from derived rings to spaces or simplicial sets: Spec C(R) = the $\infty$-groupoid of tensor functors from C to R-modules. In the case that C is the infinity-categorical form of QCoh (X) for X a scheme this recovers X, giving a result refining Balmer's (this was worked out by Nadler, Francis and myself, and many others I think -- in particular Toen and Lurie understood this long ago). I feel strongly this is a better point of view than trying to define a locally ringed space from a triangulated category, and has more of a chance to extend to stacks. In particular C tautologically sheafifies (as a sheaf of symmetric monoidal $\infty$-categories) over X.

Things get much more interesting though in the case that C=Qcoh X for X a stack (with affine diagonal, or everything fails immediately) -- eg for C=Reps(G)=QCoh (BG), the setting of usual Tannakian formalism. There one quickly learns that the above naive picture is false. Namely take G to be just the multiplicative group $G_m$. Then there are a lot of interesting fiber functors on the derived Rep G = complexes of graded vector spaces that are not given by points of BG_m (ie are not the usual "forgetful" fiber functor) --- namely we can send the defining one-dim rep of $G_m$ not to the one-dim vector space in degree zero, but in any degree we want (ie apply shift) - since this is a generator it determines the rest of the fiber functor. This proves that a naive Tannakian theorem fails in the derived setting, without imposing some "connectivity" hypotheses (ie without having t-structures around). I believe a student of Toen's is writing a thesis on this subject, but I don't know the precise statements (and so won't "out" his results).

Of course it's a little disappointing to need BOTH a tensor structure and a t-structure, since morally either one should be enough for reconstruction (having a t-structure means we can recover the abelian category of quasicoherent sheaves, which by Rosenberg recovers X already in the case of schemes without need for tensor structure). But for general stacks I don't know of a better answer.

The Tannakian reconstruction for schemes and more generally, geometric stacks from QCoh(X) with its tensor structure (due to Jacob Lurie) is explained in my answer to Tannakian Formalism . One can (and one has) try to extend this to the derived setting, where one ought to get a statement much stronger than Balmer's reconstruction theorem. First of all one needs to replace triangulated categories, which are too coarse to work with effectively, with symmetric monoidal $(\infty,1)$-categories (or symm. monoidal dg categories say in characteristic zero).

There's an obvious functor from such objects C to derived stacks Spec C given by Tannakian reconstruction -- it's defined just as in the abelian case (see my answer above). Namely we build Spec C's functor of points, as a functor from derived rings to spaces or simplicial sets: Spec C(R) = the $\infty$-groupoid of tensor functors from C to R-modules. In the case that C is the infinity-categorical form of QCoh (X) for X a scheme this recovers X, giving a result refining Balmer's (this was worked out by Nadler, Francis and myself, and many others I think -- in particular Toen and Lurie understood this long ago). I feel strongly this is a better point of view than trying to define a locally ringed space from a triangulated category, and has more of a chance to extend to stacks. In particular C tautologically sheafifies (as a sheaf of symmetric monoidal $\infty$-categories) over X.

Things get much more interesting though in the case that C=Qcoh X for X a stack (with affine diagonal, or everything fails immediately) -- eg for C=Reps(G)=QCoh (BG), the setting of usual Tannakian formalism. There one quickly learns that the above naive picture is false. Namely take G to be just the multiplicative group $G_m$. Then there are a lot of interesting fiber functors on the derived Rep G = complexes of graded vector spaces that are not given by points of BG_m (ie are not the usual "forgetful" fiber functor) --- namely we can send the defining one-dim rep of $G_m$ not to the one-dim vector space in degree zero, but in any degree we want (ie apply shift) - since this is a generator it determines the rest of the fiber functor. This proves that a naive Tannakian theorem fails in the derived setting, without imposing some "connectivity" hypotheses (ie without having t-structures around). I believe a student of Toen's is writing a thesis on this subject, but I don't know the precise statements (and so won't "out" his results).

Of course it's a little disappointing to need BOTH a tensor structure and a t-structure, since morally either one should be enough for reconstruction (having a t-structure means we can recover the abelian category of quasicoherent sheaves, which by Rosenberg recovers X already in the case of schemes without need for tensor structure). But for general stacks I don't know of a better answer.