We propose an adaptive inexact version of a class of semismooth Newton methods. As a model problem, we study the system of variational inequalities describing the contact between two membranes. This problem is written using a complementary formulation and discretized with conforming finite elements, yielding a nonlinear, semismooth (non-differentiable) algebraic system. We consider any iterative linearization algorithm like the Newton-min or the Newton Fischer--Burmeister which we complement by any iterative linear algebraic solver. We then derive an a posteriori estimate on the error between the exact solution and the approximate solution which is valid on any step of the linearization and algebraic resolutions. Our estimate is based on flux reconstructions in discrete subspaces of $\mathbf{H}(\mathrm{div}, \Omega)$ and on potential reconstructions in discrete subspaces of $H^1(\Omega)$ satisfying the constraints. It distinguishes the discretization, linearization, and algebraic components of the error. Consequently, we can formulate adaptive stopping criteria for both solvers, giving rise to an adaptive version of the considered inexact semismooth Newton algorithm. Under these criteria, the efficiency of our estimates is also established, meaning that we prove them equivalent with the error up to a generic constant. Numerical experiments for the Newton-min algorithm in combination with the GMRES algebraic solver confirm the efficiency of the developed adaptive method.