For the focusing energy critical wave equation in 5D, we construct a solution showing the inelastic nature of the collision of any two solitons, except the special case of two solitons of same scaling and opposite signs. Beyond its own interest as one of the first rigorous studies of the collision of solitons for a non-integrable model, the case of the quartic gKdV equation being partially treated by the authors in previous works, this result can be seen as part of a wider program aiming at establishing the soliton resolution conjecture for the critical wave equation. This conjecture has already been established in the 3D radial case and in the general case in 3, 4 and 5D along a sequence of times by Duyckaerts, Kenig and Merle. The study of the nature of the collision requires a refined approximate solution of the two-soliton problem and a precise determination of its space asymptotics. To prove inelasticity, these asymptotics are combined with the method of channels of energy.