We study the interactions of two or more solitary waves in the Adlam-Allen model describing the evolution of a (cold) plasma of positive and negative charges, in the presence of electric and transverse magnetic fields. In order to show that the interactions feature an exponentially repulsive nature, we elaborate two distinct approaches: (a) using energetic considerations and the Hamiltonian structure of the model, and (b) using the so-called Manton method. We compare these findings with results of direct simulations, and we identify adjustments necessary to achieve a quantitative match between them. Additional connections are made, such as with solitons of the Korteweg-de Vries equation. New challenges are identified in connection to this model and its solitary waves.