  We use the functional renormalization group and the $\epsilon$-expansion
concertedly to explore multicritical universality classes for coupled
$\bigoplus_i O(N_i)$ vector-field models in three Euclidean dimensions.
Exploiting the complementary strengths of these two methods we show how to make
progress in theories with large numbers of interactions, and a large number of
possible symmetry-breaking patterns. For the three- and four-field models we
find a new fixed point that arises from the mutual interaction between
different field sectors, and we establish the absence of infrared-stable fixed
point solutions for the regime of small $N_i$. Moreover, we explore these
systems as toy models for theories that are both asymptotically safe and
infrared complete. In particular, we show that these models exhibit complete
renormalization group trajectories that begin and end at nontrivial fixed
points.
