We present a method to simulate toponium formation events at the LHC using the Green's function of non-relativistic QCD in the Coulomb gauge, which governs the momentum distribution of top quarks in the presence of the QCD potential. This Green's function can be employed to re-weight any matrix elements relevant for $t\bar{t}$ production and decay processes where a colour-singlet top-antitop pair is produced in the $S$-wave at threshold. As an example, we study the formation of $\eta_t$ toponium states in the gluon fusion channel at the LHC, combining the re-weighted matrix elements with parton showering.