  The search for novel entangled phases of matter has lead to the recent
discovery of a new class of ``entanglement transitions'', exemplified by random
tensor networks and monitored quantum circuits. Most known examples can be
understood as some classical ordering transitions in an underlying statistical
mechanics model, where entanglement maps onto the free energy cost of inserting
a domain wall. In this paper, we study the possibility of entanglement
transitions driven by physics beyond such statistical mechanics mappings.
Motivated by recent applications of neural network-inspired variational
Ans\"atze, we investigate under what conditions on the variational parameters
these Ans\"atze can capture an entanglement transition. We study the
entanglement scaling of short-range restricted Boltzmann machine (RBM) quantum
states with random phases. For uncorrelated random phases, we analytically
demonstrate the absence of an entanglement transition and reveal subtle finite
size effects in finite size numerical simulations. Introducing phases with
correlations decaying as $1/r^\alpha$ in real space, we observe three regions
with a different scaling of entanglement entropy depending on the exponent
$\alpha$. We study the nature of the transition between these regions, finding
numerical evidence for critical behavior. Our work establishes the presence of
long-range correlated phases in RBM-based wave functions as a required
ingredient for entanglement transitions.
