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Context. Globular clusters (GCs) are fundamental tools to unveil the interaction and merger history of their host galaxies. Aims. Our goal is to perform the photometric analysis of the globular cluster system (GCS) of the elliptical galaxy NGC 4709, which is the brightest galaxy of the Cen 45 spiral-rich galaxy group, and to highlight its interaction history with NGC 4696, the giant elliptical galaxy of the Cen 30 subcluster. Methods. We obtained deep Magellan 6.5 m/MegaCam (g′ r′ i′) photometry, with which we identified a sample of 556 GC candidates around NGC 4709 that were analyzed in the context of the interaction history with the giant elliptical NGC 4696 and other galaxies of the Cen 45 group. After modeling and subtracting the galaxy light, we used criteria based on shape, colors and magnitude to select GC candidates. Results. Our results point toward a complex interaction history that shaped the GCS of NGC 4709. Inside a galactocentric radius r < 5× reff, two populations were found with mean colors of (g′ i′)0 = 0.905 ± 0.009 mag and (g′- i′)0 = 1.170 ± 0.008 mag. The azimuthal distribution of the GCs show peaks at the position angles PA1 = 92 and PA2 = 293, with PA2 coinciding with the direction toward NGC 4696, confirming that the interaction between these galaxies shaped the GCS of NGC 4709. From the GC luminosity function we derived a distance of 29.9 ± 2.1 Mpc, which is much closer than the other galaxies of the Centaurus cluster, and a specific frequency of SN = 3.7 ± 0.5, in good agreement with previously estimated values. From the GCs density maps, we identified overdensities corresponding to the positions of five other galaxies of the Centaurus cluster, and we found a bridge of GCs between NGC 4709 and NGC 4696 with a distance of d = 34.69 ± 2.21 Mpc, which is between that of the two galaxies. Conclusions. All of these findings point toward a complex GCS for NGC 4709, strongly influenced by the interaction with NGC 4696, and confirm previous findings that the galaxy's apparent distance is smaller than that of the main cluster galaxy NGC 4696 by ∼8.5 Mpc and that it is smaller than the distance of the other Centaurus' galaxies, making it an outlier in Centaurus and suggesting a past first encounter with Cen 30. In future work, we will combine our data with E-MOSAICS simulations to unravel the trajectory of NGC 4709, from the past interaction to the future new rendezvous with NGC 4696.