We study the ground-state phase diagram of two-dimensional two-component (or pseudospin-1/2) Bose gases in a high synthetic magnetic field in the space of the total filling factor and the ratio of the intercomponent coupling $g_{uparrowdownarrow}$ to the intracomponent one $g>0$. Using exact diagonalization, we find that when the intercomponent coupling is attractive ($g_{uparrowdownarrow}<0$), the product states of a pair of nearly uncorrelated quantum Hall states are remarkably robust and persist even when $|g_{uparrowdownarrow}|$ is close to $g$. This contrasts with the case of an intercomponent repulsion, where a variety of spin-singlet quantum Hall states with high intercomponent entanglement emerge for $g_{uparrowdownarrow}approx g$. We interpret this marked dependence on the sign of $g_{uparrowdownarrow}$ in light of pseudopotentials on a sphere, and also explain recent numerical results in two-component Bose gases in mutually antiparallel magnetic fields where a qualitatively opposite dependence on the sign of $g_{uparrowdownarrow}$ is found. Our results thus unveil an intriguing connection between multicomponent quantum Hall systems and quantum spin Hall systems in minimal setups.
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