  Let $K$ be a reductive subgroup of a reductive group $G$ over an
algebraically closed field $k$. The notion of relative complete reducibility,
introduced in previous work of Bate-Martin-Roehrle-Tange, gives a purely
algebraic description of the closed $K$-orbits in $G^n$, where $K$ acts by
simultaneous conjugation on $n$-tuples of elements from $G$. This extends work
of Richardson and is also a natural generalization of Serre's notion of
$G$-complete reducibility. In this paper we revisit this idea, giving a
characterization of relative $G$-complete reducibility which directly
generalizes equivalent formulations of $G$-complete reducibility. If the
ambient group $G$ is a general linear group, this characterization yields
representation-theoretic criteria. Along the way, we extend and generalize
several results from the aforementioned work of Bate-Martin-Roehrle-Tange.
