expression, godan verb (-su)
cry tears of joy; weep for joy
A set phrase meaning to shed tears from deep joy or gratitude, often in a religious or deeply emotional context. The word 随喜 (ずいき) is a Buddhist term for rejoicing in others' good deeds.
彼女は合格の知らせを聞いて、随喜の涙を流した。
She cried tears of joy when she heard the news of her acceptance.
嬉し涙 is a more common, everyday word for tears of joy, while 随喜の涙を流す is a literary, set phrase with Buddhist origins.
From 随喜 (ずいき, 'rejoicing in others' good deeds' in Buddhism) + の + 涙 (なみだ, 'tears') + を流す (ながす, 'to shed'). The phrase is rooted in Buddhist terminology.