  A model of two-species bosons moving on the sites of a lattice is studied at
nonzero temperature, focusing on magnetic order and superfluid-insulator
transitions. Firstly, Landau theory is used to find the general structure of
the phase diagram, and in particular to demonstrate the presence of first-order
transitions and hysteresis in the vicinity of a multicritical point. Secondly,
an explicit thermodynamic phase diagram is calculated using an approach based
on a field-theoretical description of the Bose-Hubbard model, which
incorporates the crucial effects of particle-number fluctuations. The maximum
transition temperature to a magnetically ordered Mott insulator is found to be
limited by the presence of the superfluid phase.
