Objective. Magnetic Particle Imaging (MPI) is an emerging medical tomographic imaging modality that enables real-time imaging with high sensitivity and high spatial and temporal resolution. For the system matrix reconstruction method, the MPI reconstruction problem is an ill-posed inverse problem that is commonly solved using the Kaczmarz algorithm. However, the high computation time of the Kaczmarz algorithm, which restricts MPI reconstruction speed, has limited the development of potential clinical applications for real-time MPI. In order to achieve fast reconstruction in real-time MPI, we propose a greedy regularized block Kaczmarz method (GRBK) which accelerates MPI reconstruction. Approach. GRBK is composed of a greedy partition strategy for the system matrix, which enables preprocessing of the system matrix into well-conditioned blocks to facilitate the convergence of the block Kaczmarz algorithm, and a regularized block Kaczmarz algorithm, which enables fast and accurate MPI image reconstruction at the same time. Main results. We quantitatively evaluated our GRBK using simulation data from three phantoms at 20dB, 30dB, and 40dB noise levels. The results showed that GRBK can improve reconstruction speed by single orders of magnitude compared to the prevalent regularized Kaczmarz algorithm including Tikhonov regularization, the non-negative Fused Lasso, and wavelet-based sparse model. We also evaluated our method on OpenMPIData, which is real MPI data. The results showed that our GRBK is better suited for real-time MPI reconstruction than current state-of-the-art reconstruction algorithms in terms of reconstruction speed as well as image quality. Significance. Our proposed method is expected to be the preferred choice for potential applications of real-time MPI.