  We compute the finite-temperature phase diagram of a pseudospin-$1/2$ Bose
gas with contact interactions, using two complementary methods: the random
phase approximation (RPA) and self-consistent Hartree-Fock theory. We show that
the inter-spin interactions, which break the (pseudo) spin-rotational symmetry
of the Hamiltonian, generally lead to the appearance of a magnetically ordered
phase at temperatures above the superfluid transition. In three dimensions, we
predict a normal easy-axis/easy-plane ferromagnet for sufficiently strong
repulsive/attractive inter-species interactions respectively. The normal
easy-axis ferromagnet is the bosonic analog of Stoner ferromagnetism known in
electronic systems. For the case of inter-spin attraction, we also discuss the
possibility of a \textit{bosonic} analog of the Cooper paired phase. This state
is shown to significantly lose in energy to the transverse ferromagnet in three
dimensions, but is more energetically competitive in lower dimensions.
Extending our calculations to a spin-orbit-coupled Bose gas with equal Rashba
and Dresselhaus-type couplings (as recently realized in experiment), we
investigate the possibility of stripe ordering in the normal phase. Within our
approximations however, we do not find an instability towards stripe formation,
suggesting that the stripe order melts below the condensation temperature,
which is consistent with the experimental observations of Ji \textit{et al.}
[Ji \textit{et al.}, Nature Physics \textbf{10}, 314 (2014)].
