  Motivated by experiments in bosonic mixtures composed of a single element in
two different hyperfine states, we study bosonic binary mixtures in the
presence of Josephson interactions between species. We focus on a particular
model with $O(2)$ isospin symmetry, lifted by an imbalanced population
parametrized by a Rabi frequency, $\Omega _{R}$, and a detuning, $\nu $, which
couples the phases of both species. We have studied the model at mean-field
approximation plus Gaussian fluctuations. We have found that both species
simultaneously condensate below a critical temperature $T_{c}$ and the relative
phases are locked by the applied laser phase, $\alpha$. Moreover, the
condensate fractions are strongly dependent on the ratio $\Omega _{R}/|\nu |$
that is not affected by thermal fluctuations.
