  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.
