  We describe multiquark clusters in quark matter within a Beth-Uhlenbeck
approach in a background gluon field coupled to the underlying chiral quark
dynamics using the Polyakov gauge which establishes the center symmetry of
color SU(3) that suppresses colored states as an aspect of confinement. Quark
confinement is modeled by a large quark mass in vacuum motivated by a confining
density functional approach. A multiquark cluster containing $n$ quarks and
antiquarks is described as a binary composite of smaller subclusters $n_1$ and
$n_2$ ($n_1+n_2=n$). It has a spectrum consisting of a bound state and a
scattering state continuum. For the corresponding cluster-cluster phase shifts
we discuss simple ans\"atze that capture the Mott dissociation of clusters as a
function of temperature and chemical potential. We go beyond the simple
"step-up-step-down" model that ignores continuum correlations and introduce an
improved model that includes them in a generic form. In order to explain the
model, we restrict ourselves here to the cases where the cluster size is $1 \le
n \le 6$. A striking result is the suppression of the abundance of colored
multiquark clusters at low temperatures by the coupling to the Polyakov loop
and their importance for a quantitative description of lattice QCD
thermodynamics at non-vanishing baryochemical potentials. An important
ingredient are Polyakov-loop generalized distribution functions of $n$-quark
clusters which are derived here for the first time. Within our approach we
calculate thermodynamic properties such as baryon density and entropy. We
demonstrate that the limits of a hadron resonance gas at low temperatures and
$\mathcal{O}(g^2)$ perturbative QCD at high temperatures are correctly
reproduced. A comparison with lattice calculations shows that our model is able
to give a unified, systematic approach to describe properties of the
quark-gluon-hadron system.
