  Recent experiments confirmed that fluctuations beyond the mean-field
approximation can lead to self-bound liquid droplets of ultra-dilute binary
Bose mixtures. We proceed beyond the beyond-mean-field approximation, and study
liquid Bose mixtures using the variational hypernetted-chain Euler Lagrange
method, which accounts for correlations non-perturbatively. 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 $\rho_1/\rho_2$ and near the total
energy minimum. Deviations from a universal dependence on the s-wave scattering
lengths are significant despite the low density.
