  The equation of state with light clusters for nuclear and stellar matter is
determined using chemical equilibrium constants evaluated from the analysis of
the recently published (Xe$+$Sn) heavy ion data, corresponding to three
reactions with different isotopic contents of the emission source. The measured
multiplicities are used to extract the thermodynamic properties, and an
in-medium correction to the ideal gas internal partition function of the
clusters is included in the analysis. This in-medium correction and its
respective uncertainty are calculated via a Bayesian analysis, with the unique
hypothesis that the different nuclear species in a given sample must correspond
to a unique common value for the density of the expanding source. Different
parameter sets for the correction are tested, and the effect of the radius of
the clusters on the thermodynamics and on the chemical equilibrium constants is
also addressed. It is shown that the equilibrium constants obtained are almost
independent of the isospin content of the analysed systems. Finally, a
comparison with a relativistic mean field model proves that data are consistent
with a universal in-medium correction of the scalar $\sigma$-meson coupling for
nucleons bound in clusters. The obtained value, $g_s/g_s^0 = 0.92 \pm 0.02$, is
larger than that obtained in a previous study not including in-medium effects
in the data analysis. This result implies a smaller effect on the binding
energy of the clusters and, as a consequence, larger melting densities, and an
increased cluster contribution in supernova matter.
