Singular behaviors are increasingly being found in structures supporting bound states in the continuum (BICs), while the nonreciprocity with spectral phase singularity has not been reported. Here, we demonstrate the origin, evolution, and application of topological phase singularity pairs (TPSPs) resulting from BICs in nonreciprocal and non-Hermitian systems. The nonreciprocity contributes to creating accidental BICs asymmetrically, each of which can split into TPSP with topological charges in reflection phases by inserting loss. The formation, annihilation, and revival processes of these TPSPs can be selectively controlled via either material or radiative loss. The criteria to predict both number and angular positions of asymmetric BICs are established. Near-complete violation of Kirchhoff’s law of thermal radiation has been correspondingly manifested over a wide angular range. The unveiled physics synergizing nonreciprocity and topology will bring new opportunities in multidisciplinary areas like thermal science, magneto-optics, or topological metasurfaces.