  Local constraints play an important role in the effective description of many
quantum systems. Their impact on dynamics and entanglement thermalization are
just beginning to be unravelled. We develop a large $N$ diagrammatic formalism
to exactly evaluate the bipartite entanglement of random pure states in large
constrained Hilbert spaces. The resulting entanglement spectra may be
classified into `phases' depending on their singularities. Our closed solution
for the spectra in the simplest class of constraints reveals a non-trivial
phase diagram with a Marchenko-Pastur (MP) phase which terminates in a critical
point with new singularities. The much studied Rydberg-blockaded/Fibonacci
chain lies in the MP phase with a modified Page correction to the entanglement
entropy, $\Delta S_1 = 0.513595\cdots$. Our results predict the entanglement of
infinite temperature eigenstates in thermalizing constrained systems and
provide a baseline for numerical studies.
