Nonlinear model predictive control (NMPC) requires the solution of a dynamic optimization problem at each sampling instant under strict timing constraints, involving nonlinear dynamics that can often be stiff or implicitly defined. The real-time iteration (RTI) scheme has been shown to allow real-world embedded applications of NMPC. The present paper proposes an extension of the standard RTI algorithm with a block-structured quasi-Newton method to obtain low-rank Jacobian updates that preserve the block structure of the optimal control problem. In addition, a particular structure-exploiting implementation is presented for implicit integration schemes such that no Jacobian evaluation is needed neither any matrix factorization. Based on a proof of concept implementation in C code, the computational performance of the algorithm is illustrated for multiple NMPC case studies.