A numerical investigation of collisions of kink-anti-kink (KK) pairs in an easy-plane classical antiferromagnetic chain is presented here, for the case of an applied field within the easy plane. An approximate kink profile as obtained from an earlier ansatz was used as the initial condition for numerical integration on a discrete lattice. As a function of the applied magnetic field and kink ansatz parameter theta _A, which measures the tilt of the spins out of the easy plane, the authors have found distinct parameter regimes resulting in transmission, reflection, and annihilation of the KK pair. Results for both the in-plane (XY) and out-of-plane (YZ) kinks are summarised, and comparison is made to KK scattering in the easy-plane ferromagnet. Similarly to the case of the easy-plane ferromagnetic CsNiF₃, it is found that these results imply difficulties in the interpretation of experiments on tetramethylammonium manganese trichloride (TMMC) in terms of classical soliton theory. For example, for fields from approximately 20 to 80 kG, the low-velocity XY collisions generally result in pair annihilation to spin waves. For fields less than 20 kG, however, the low-velocity KK XY pair transmit. Experimental results for TMMC are discussed in the light of these features.