  Suppose $V^G$ is the fixed-point vertex operator subalgebra of a compact
group $G$ acting on a simple abelian intertwining algebra $V$. We show that if
all irreducible $V^G$-modules contained in $V$ live in some braided tensor
category of $V^G$-modules, then they generate a tensor subcategory equivalent
to the category $\mathrm{Rep}\,G$ of finite-dimensional representations of $G$,
with associativity and braiding isomorphisms modified by the abelian
$3$-cocycle defining the abelian intertwining algebra structure on $V$.
Additionally, we show that if the fusion rules for the irreducible
$V^G$-modules contained in $V$ agree with the dimensions of spaces of
intertwiners among $G$-modules, then the irreducibles contained in $V$ already
generate a braided tensor category of $V^G$-modules. These results do not
require rigidity on any tensor category of $V^G$-modules and thus apply to many
examples where braided tensor category structure is known to exist but rigidity
is not known; for example they apply when $V^G$ is $C_2$-cofinite but not
necessarily rational. When $V^G$ is both $C_2$-cofinite and rational and $V$ is
a vertex operator algebra, we use the equivalence between $\mathrm{Rep}\,G$ and
the corresponding subcategory of $V^G$-modules to show that $V$ is also
rational. As another application, we show that a certain category of modules
for the Virasoro algebra at central charge $1$ admits a braided tensor category
structure equivalent to $\mathrm{Rep}\,SU(2)$, up to modification by an abelian
$3$-cocycle.
