Using quantum Monte Carlo simulations we have mapped out the zero-temperature phase diagram of a symmetric electron-hole bilayer with twofold valley degeneracy, as a function of the interlayer distance $d$ and in-layer density $n$. We find that the effect of the valley degeneracy is to shrink the region of stability of the excitonic condensate, in favor of quadriexcitons at small $d$ and of the four-component plasma at large $d$, with minor effects on the value of the excitonic condensate fraction. The enclosure of the condensate in a density window possibly explains why anomalous tunneling conductivity, interpreted as a signature of condensation, is observed only between two finite values of carrier density in graphene bilayers. Our phase diagram may provide directions to select device parameters for future experiments.

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• Superfluidity
• Bilayer graphene
• Two-dimensional electron system
• Diffusion quantum Monte Carlo
• Quantum Monte Carlo