We study a real-time iteration (RTI) scheme for solving online optimization problem appeared in nonlinear optimal control. The proposed RTI scheme modifies the existing RTI-based model predictive control (MPC) algorithm, by selecting the stepsize of each Newton step at each sampling time using a differentiable exact augmented Lagrangian. The scheme can adaptively select the penalty parameters of augmented Lagrangian on the fly, which are shown to be stabilized after certain time periods. We prove under generic assumptions that, by involving stepsize selection instead of always using a full Newton step (like what most of the existing RTIs do), the scheme converges globally: for any initial point, the KKT residuals of the subproblems converge to zero. A key step is to show that augmented Lagrangian keeps decreasing as horizon moves forward. We demonstrate the global convergence behavior of the proposed RTI scheme in a numerical experiment.