  This manuscript aims to establish the gravitational junction conditions(JCs)
for the $f(\mathcal{G},~T)$ gravity. In this gravitational theory, $f$ is an
arbitrary function of Gauss-Bonnet invariant $\mathcal{G}$ and the trace of the
energy-momentum tensor $T_{\mu\nu}$ i.e., $T$. We start by introducing this
gravity theory in its usual geometrical representation and posteriorly obtain a
dynamically equivalent scalar-tensor demonstration on which the arbitrary
dependence on the generic function $f$ in both $\mathcal G$ and $T$ is
exchanged by two scalar fields and scalar potential. We then derive the JCs for
matching between two different space-times across a separation hyper-surface
$\Sigma$, assuming the matter sector to be described by an isotropic perfect
fluid configuration. We take the general approach assuming the possibility of a
thin-shell arising at $\Sigma$ between the two space-times. However, our
results establish that, for the distribution formalism to be well-defined,
thin-shells are not allowed to emerge in the general version of this theory. We
thus obtain instead a complete set of JCs for a smooth matching at $\Sigma$
under the same conditions. The same results are then obtained in the
scalar-tensor representation of the theory, thus emphasizing the equivalence
between these two representations. Our results significantly constrain the
possibility of developing models for alternative compact structures supported
by thin-shells in $f(\mathcal{G},~T)$ gravity, e.g. gravastars and thin-shell
wormholes, but provide a suitable framework for the search of models presenting
a smooth matching at their surface, from which perfect fluid stars are possible
examples.
