  We study the probability distribution of entanglement in the Quantum
Symmetric Simple Exclusion Process, a model of fermions hopping with random
Brownian amplitudes between neighboring sites in one dimension. We consider a
protocol where the system is initialized in a pure product state of $M$
particles, and focus on the late-time distribution of R\'enyi-$q$ entropies for
a subsystem of size $\ell$. By means of a Coulomb gas approach from Random
Matrix Theory, we compute analytically the large-deviation function of the
entropy in the thermodynamic limit. For $q>1$, we show that, depending on the
value of the ratio $\ell/M$, the entropy distribution displays either two or
three distinct phases, ranging from low- to high-entanglement regimes. These
phases are connected by third-order critical points, which can be understood in
terms of a transition in the corresponding charge density of the Coulomb gas.
Our analytic results are supported by numerical Monte Carlo simulations.
