The formal spectrum of a tensor-triangulated category

To any essentially small tensor-triangulated category $\mathcal{K}$ and Thomason subset $Y\subseteq \mathrm{S}\mathrm{p}\mathrm{c}(\mathcal{K})$ we associate a ringed space $(\mathrm{S}\mathrm{p}\mathrm{f}(\mathcal{K},Y),{\mathcal{O}}_{\mathrm{S}\mathrm{p}\mathrm{f}(\mathcal{K},Y)}),$ called the formal spectrum of $(\mathcal{K},Y)$. We establish basic properties of this construction and compute it in several examples from algebraic geometry, chromatic homotopy theory, equivariant homotopy theory, and modular representation theory.