  We study models with three coupled vector fields characterized by
$O(N_1)\oplus O(N_2) \oplus O(N_3)$ symmetry. Using the nonperturbative
functional renormalization group, we derive $\beta$ functions for the couplings
and anomalous dimensions in $d$ dimensions. Specializing to the case of three
dimensions, we explore interacting fixed points that generalize the $O(N)$
Wilson-Fisher fixed point. We find a symmetry-enhanced isotropic fixed point, a
large class of fixed points with partial symmetry enhancement, as well as
partially and fully decoupled fixed point solutions. We discuss their stability
properties for all values of $N_1, N_2$, and $N_3$, emphasizing important
differences to the related two-field models. For small numbers of field
components we find no stable fixed point solutions, and we argue that this can
be attributed to the presence of a large class of possible (mixed) couplings in
the three-field and multifield models. Furthermore, we contrast different
mechanisms for stability interchange between fixed points in the case of the
two- and three-field models, which generically proceed through fixed-point
collisions.
