  We study the braided tensor structure of line operators in the topological A
and B twists of abelian 3d $\mathcal{N}=4$ gauge theories, as accessed via
boundary vertex operator algebras (VOA's). We focus exclusively on abelian
theories. We first find a non-perturbative completion of boundary VOA's in the
B twist, which start out as certain affine Lie superalebras; and we construct
free-field realizations of both A and B-twist VOA's, finding an interesting
interplay with the symmetry fractionalization group of bulk theories. We use
the free-field realizations to establish an isomorphism between A and B VOA's
related by 3d mirror symmetry. Turning to line operators, we extend previous
physical classifications of line operators to include new monodromy defects and
bound states. We also outline a mechanism by which continuous global symmetries
in a physical theory are promoted to higher symmetries in a topological twist
-- in our case, these are infinite one-form symmetries, related to boundary
spectral flow, which structure the categories of lines and control abelian
gauging. Finally, we establish the existence of braided tensor structure on
categories of line operators, viewed as non-semisimple categories of modules
for boundary VOA's. In the A twist, we obtain the categories by extending
modules of symplectic boson VOA's, corresponding to gauging free
hypermultiplets; in the B twist, we instead extend Kazhdan-Lusztig categories
for affine Lie superalgebras. We prove braided tensor equivalences among the
categories of 3d-mirror theories. All results on VOA's and their module
categories are mathematically rigorous; they rely strongly on recently
developed techniques to access non-semisimple extensions.
