In this paper, a novel optimality-tracking algorithm for solving Economic Nonlinear Model Predictive Control (ENMPC) problems in real-time is presented. Developing online schemes for ENMPC is challenging, since it is unclear how convexity of the Quadratic Programming (QP) problem, which is obtained by linearisation of the NMPC program around the current iterate, can be enforced efficiently. Therefore, we propose addressing the problem by means of an augmented Lagrangian formulation. Our tracking scheme consists of a fixed number of inexact Newton steps computed on an augmented Lagrangian subproblem followed by a dual update per time step. Under mild assumptions on the number of iterations and the penalty parameter, it can be proven that the sub-optimality error provided by the parametric algorithm remains bounded over time. This result extends the authors' previous works from a theoretical and a computational perspective. Efficacy of the approach is demonstrated on an ENMPC example consisting of a bioreactor.