Despite numerous recent advances in medical ultrasound imaging, reverberation clutter from near-field anatomical structures, such as the abdominal wall, ribs, and tissue layers, is one of the major sources of ultrasound image quality degradation. Reverberation clutter signals are undesirable echoes, which arise as a result of multiple reflections of acoustic waves between the boundaries of these structures, and cause fill-in to lower image contrast. In order to mitigate the undesirable reverberation clutter effects, we present, in this paper, a new beamforming technique called multiphase apodization with cross correlation (MPAX), which is an improved version of our previous technique, dual apodization with cross correlation (DAX). While DAX uses a single pair of complementary amplitude apodizations, MPAX utilizes multiple pairs of complementary sinusoidal phase apodizations to intentionally introduce grating lobes from which an improved weighting matrix can be produced to effectively suppress reverberation clutter. Our experimental sponge phantom and preliminary in vivo results from human subjects presented in this paper suggest that MPAX is a highly effective technique in suppressing reverberation clutter and has great potential for producing high contrast ultrasound images for more accurate diagnosis in clinics.