Recently, different real-time optimization (RTO) schemes that guarantee feasibility of all RTO iterates and monotonic convergence to the optimal plant operating point have been proposed. However, simulations reveal that these schemes converge very slowly to the plant optimum, which may be prohibitive in applications. This note proposes an RTO scheme based on second-order surrogate models of the objective and the constraints, which enforces feasibility of all RTO iterates, i.e., plant constraints are satisfied at all iterations. In order to speed up convergence, we suggest an online adaptation strategy of the surrogate models that is based on trust-region ideas. The efficacy of the proposed RTO scheme is demonstrated in simulations via both a numerical example and the steady-state optimization of the Williams-Otto reactor.