We construct and study a nonstandard t-structure on the derived category of equivariant coherent sheaves on the Braverman-Finkelberg-Nakajima space of triples $\mathcal{R}_{G,N}$, where $N$ is a representation of a reductive group $G$. Its heart $\mathcal{KP}_{G,N}$ is a finite-length, rigid, monoidal abelian category with renormalized $r$-matrices. We refer to objects of $\mathcal{KP}_{G,N}$ as Koszul-perverse coherent sheaves. Simple objects of $\mathcal{KP}_{G,N}$ define a canonical basis in the quantized $K$-theoretic Coulomb branch of the associated gauge theory. These simples possess various characteristic properties of Wilson-'t Hooft lines, and we interpret our construction as an algebro-geometric definition of the category of half-BPS line defects in a 4d $\mathcal{N}=2$ gauge theory of cotangent type.