This paper proposes a real-time algorithm for processing and quality improvement of synchrophasor data (SD). The proposed algorithm first recovers the missing SD reported by phasor measurement units (PMUs), and performs low-rank approximation on data streams. Then, the enhanced data stream can be redirected toward various power system applications. The nonconvex matrix completion (MC) method with Schatten-q quasi-norm (lq) penalty is used to recover the missing SD in real-time. Unlike most MC methods which have been developed for batch data processing, the proposed method is able to perform fast recovery of streaming data even for high reporting rates of PMU data. The low-rank approximation method is used to suppress the noise of the streaming SD, and to efficiently compress the batch data for archiving. Real-life PMU data as well as simulation data are used to evaluate the performance of the proposed algorithms. The results obtained using both real experimental and simulation SD confirm that the proposed SD processing framework significantly improves the quality of data, particularly during transient conditions and in noisy environments.