  We develop an approach to the study of Coulomb branch operators in 3D
$\mathcal{N}=4$ gauge theories and the associated quantization structure of
their Coulomb branches. This structure is encoded in a one-dimensional TQFT
subsector of the full 3D theory, which we describe by combining several
techniques and ideas. The answer takes the form of an associative and
noncommutative star product algebra on the Coulomb branch. For `good' and
`ugly' theories (according to the Gaiotto-Witten classification), we also
exhibit a trace map on this algebra, which allows for the computation of
correlation functions and, in particular, guarantees that the star product
satisfies a truncation condition. This work extends previous work on abelian
theories to the non-abelian case by quantifying the monopole bubbling that
describes screening of GNO boundary conditions. In our approach, monopole
bubbling is determined from the algebraic consistency of the OPE. This also
yields a physical proof of the Bullimore-Dimofte-Gaiotto abelianization
description of the Coulomb branch.
