Using data collected at the Pierre Auger Observatory, we search for signatures of instanton-induced processes that would provide evidence of super-heavy particles decaying in the Galactic halo. Such particles could have been produced sufficiently during the post-inflationary epoch to match the relic abundance of dark matter inferred today. The non-observation of these signatures allows us to probe the instanton strength and to derive a bound, the best ever obtained from instanton-mediated processes, on the reduced coupling constant of gauge interactions in the dark sector: $\alpha_X \lesssim 0.09$, for $10^{9} \lesssim M_X/{\rm GeV} < 10^{19}$.