  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
$g_{\uparrow\downarrow}$ to the intracomponent one $g>0$. Using exact
diagonalization, we find that when the intercomponent coupling is attractive
($g_{\uparrow\downarrow}<0$), the product states of a pair of nearly
uncorrelated quantum Hall states are remarkably robust and persist even when
$|g_{\uparrow\downarrow}|$ 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 $g_{\uparrow\downarrow}\approx
g$. We interpret this marked dependence on the sign of $g_{\uparrow\downarrow}$
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
$g_{\uparrow\downarrow}$ is found. Our results thus unveil an intriguing
connection between multicomponent quantum Hall systems and quantum spin Hall
systems in minimal setups.
