Optofluidic time-stretch microscopy is a powerful tool in imaging flow cytometry as it enables continuous image acquisition at a frame rate higher than 10,000 frames per second. In addition to bright-field imaging that provides morphological information, attempts have been made to integrate quantitative phase imaging (QPI) with optofluidic time-stretch microscopy to acquire information related to subcellular structure, such as the refractive index and thickness. However, the applicability of such methods is hindered by errors introduced during phase unwrapping and the need for a high-bandwidth photodetector. To overcome these limitations, here we demonstrate optofluidic time-stretch QPI based on an acousto-optic modulator (AOM) that acquires intensity and phase image with a low-bandwidth photodetector without phase-unwrapping errors. In our system, the signal beam that carries cellular information interferes with the reference beam, the frequency of which is shifted by 1/4 of the repetition frequency of the laser by an AOM. The beat note is then detected by a normal photodetector, and its waveform that consists of groups of four successive pulses is converted into phase and intensity images with simple calculations. Therefore, we lower the requirement of the photodetector bandwidth and eliminate the errors in phase unwrapping while maintaining a throughput of 10,000 cells per second. These advantages of our system offer new possibilities for high-throughput label-free cancer cell detection in blood by looking at cellular phase information including structural features, enabling early cancer detection and improving the effectiveness of treatment.