In this talk, we explain how to equip the Balmer spectrum of a rigidly-compactly generated symmetric monoidal stable infinity-category with a natural structure sheaf, generalizing gluing techniques of Balmer-Favi in a systematic fashion. Following tensor-triangular philosophy in porting over statements from algebraic geometry, we provide a universal property for this equipment (in the style of an affine scheme). As an application, we explain how to recover the computation of the thick tensor ideals of the perfect complexes on a qcqs scheme, in addition to partially generalizing the statement to nice classes of spectral stacks. In another direction, we explain how one might utilise these techniques to import methods from spectral algebraic geometry into a wider variety of contexts.