While agunot are discussed extensively in rabbinic literature, agunim (i.e., male “agunot”) rarely are. I found one such discussion, though, tucked in an early American responsa sefer I recently acquired.
This one-page printed document from 1925 is a psak din from a beit din in Brownsville, Brooklyn under the leadership of R. Moses Benjamin Tomashoff (d. 1960). Originally from Slutzk, Tomashoff moved to Brownsville (then called the “Jerusalem of America”) and was an active member of the Agudas Harabonim and editor of the rabbinic journal Yagdil Torah.
The psak din concerned a woman who was declared insane 13 years prior and under the care of Central Islip Psychiatric Center, one of the largest psychiatric hospitals in New York. (In its final years – it closed in 1996 – Islip Medical Center set aside a portion of its on-grounds cemetery for Jewish residents, which today sits adjacent to the Touro Law School in Islip.)
Since this woman was not capable of receiving a get due to her mental state, her husband couldn’t marry another woman and was forced to raise his children alone. Interestingly, the beit din notes that he had no daughters to help him take care of the other children.
The beit din found a halachic basis to allow him to marry a second woman on condition that an additional 97 rabbis agreed to their ruling (which would result in a heter meah rabbanim). The beit din also stipulated that the husband commit to pay for his first wife’s medical needs perpetually and that he first obtain a civil divorce.
Presumably, this document was sent to other rabbis to sign until 100 of them agreed to let him marry another woman.