  We give a novel characterization of the Euclidean quantum field theory with
exponential interaction $\nu$ on $\mathbb{R}^2$ through a renormalized
integration by parts (IbP) formula, or otherwise said via an Euclidean
Dyson-Schwinger equation for expected values of observables. In order to obtain
the well-posedness of the singular IbP problem, we import some ideas used to
analyse singular SPDEs and we require the measure to "look like" the Gaussian
free field (GFF) in the sense that a suitable Wasserstein distance from the GFF
is finite. This guarantees the existence of a nice coupling with the GFF which
allows to control the renormalized IbP formula.
