Phases of the spherical harmonic analysis of full-sky cosmic microwave background (CMB) temperature data contain useful information complementary to the ubiquitous angular power spectrum. In this Letter we present a new method of phase analysis on incomplete sky maps. It is based on Fourier phases of equal-latitude pixel rings of the map, which are related to the mean angle of the trigonometric moments from the full-sky phases. It has an advantage for probing regions of interest without tapping polluted Galactic plane area, and can localize non-Gaussian features. This method is therefore very useful for detection of departure from Gaussianity and statistical isotropy in the CMB.