  Field theories with combinatorial non-local interactions such as tensor
invariants are interesting candidates for describing a phase transition from
discrete quantum-gravitational to continuum geometry. In the so-called
cyclic-melonic potential approximation of a tensorial field theory on the
$r$-dimensional torus it was recently shown using functional renormalization
group techniques that no such phase transition to a condensate phase with a
tentative continuum geometric interpretation is possible. Here, keeping the
same approximation, we show how to overcome this limitation amending the theory
by local degrees freedom on $\mathbb{R}^d$. We find that the effective $r-1$
dimensions of the torus part dynamically vanish along the renormalization group
flow while the $d$ local dimensions persist up to small momentum scales.
Consequently, for $d>2$ one can find a phase structure allowing also for phase
transitions.
