  We develop a framework of coupled transport equations for open heavy flavor
and quarkonium states, in order to describe their transport inside the
quark-gluon plasma. Our framework is capable of studying simultaneously both
open and hidden heavy flavor observables in heavy-ion collision experiments and
can account for both, uncorrelated and correlated recombination. Our
recombination implementation depends on real-time open heavy quark and
antiquark distributions. We carry out consistency tests to show how the
interplay among open heavy flavor transport, quarkonium dissociation and
recombination drives the system to equilibrium. We then apply our framework to
study bottomonium production in heavy-ion collisions. We include
$\Upsilon(1S)$, $\Upsilon(2S)$, $\Upsilon(3S)$, $\chi_b(1P)$ and $\chi_b(2P)$
in the framework and take feed-down contributions during the hadronic gas stage
into account. Cold nuclear matter effects are included by using nuclear parton
distribution functions for the initial primordial heavy flavor production. A
calibrated $2+1$ dimensional viscous hydrodynamics is used to describe the bulk
QCD medium. We calculate both the nuclear modification factor $R_{\mathrm{AA}}$
of all bottomonia states and the azimuthal angular anisotropy coefficient $v_2$
of the $\Upsilon(1S)$ state and find that our results agree reasonably with
experimental measurements. Our calculations indicate that correlated cross-talk
recombination is an important production mechanism of bottomonium in current
heavy-ion experiments. The importance of correlated recombination can be tested
experimentally by measuring the ratio of $R_{\mathrm{AA}}(\chi_b(1P))$ and
$R_{\mathrm{AA}}(\Upsilon(2S))$.
