  The Renormalisation Group (RG) is a systematic procedure used to regularise
divergences appearing as artefacts when constructing solutions to a large class
of differential problems, whether perturbatively or not. This paper is devoted
to performing it in the non-perturbative context of flat sections of
meromorphic connections in holomorphic principal bundles over a base complex
curve, with reductive and complex structure Lie groups. Using the
interpretation of Kunuhiro of RG in terms of envelopes of families of
solutions, we identify these envelopes as particular flat sections associated
to points on the corresponding quantum spectral curve, namely certain divisors
on the base curve with local coefficients in the adjoint bundle, yielding
integral representations. This shows that the non-perturbative renormalisation
of flat sections of principal bundles over curves fits naturally within the
description of meromorphic connections via their quantum geometry, or homology
\textit{\`a la} Goldman, where the flat sections and their deformations are
parameterized by degree zero and one chains respectively. We conclude the paper
by mentioning upcoming applications.
