  The miscibility condition for a binary mixture of two interacting
Bose-Einstein condensates is shown to be deeply affected by interaction driven
thermal fluctuations. These give rise to a first order phase transition to a
demixed phase with full spatial separation of the two condensates, even if the
mixture is miscible at zero temperature. Explicit predictions for the
isothermal compressibility, the spin susceptibility, and the phase transition
temperature $T_M$ are obtained in the framework of Popov theory, which properly
includes beyond mean-field quantum and thermal fluctuations in both the spin
and density channels. For a mixture of two sodium condensates occupying the
hyperfine states $\lvert F=1, m_F=1 \rangle$ and $\lvert F=1, m_F=-1 \rangle$
respectively, $T_M$ is predicted to occur at about $0.7$ times the usual BEC
critical temperature.
