Recent experiments confirmed that fluctuations beyond the mean-field approximation can lead to self-bound liquid droplets of ultradilute binary Bose mixtures. We proceed beyond the beyond-mean-field approximation and study liquid Bose mixtures by using the variational hypernetted-chain Euler–Lagrange method, which accounts for correlations nonperturbatively. Focusing on the case of a mixture of uniform density, as realized inside large saturated droplets, we study the conditions for stability against evaporation of one of the components (both chemical potentials need to be negative) and against liquid-gas phase separation (spinodal instability), the latter being accompanied by a vanishing speed of sound. Dilute Bose mixtures are stable only in a narrow range near an optimal ratio ¿1/¿2 and near the total energy minimum. Deviations from a universal dependence on the s-wave scattering lengths are significant despite the low density.