A vizier in the court of Jessore (now in Bangladesh) by the name of Mohammad Tahir Pir Ali is said to be the eponymous source of the term, Pirali. He was a Brahmin Hindu who converted to Islam, something that orthodox Hindus didn't much care for. As a result, they chose to shun not only Pir Ali, but all his relatives, and eventually, their descendants as well. Pirali grew to become a derogatory term used to refer to such excommunicated Hindus and over time, was also employed in reference to any Hindu families whose members had converted to Islam.
The Tagore family were descendants of one such Pirali family. The ostracism faced by them for being piralis drove Dwarkanath, Debendranath (grandfather and father of Rabindranath), and other Tagores to join the Brahmo movement in an attempt to rid Hindu society of its bigotries.