noun
A formal, often deadly, fight to settle a dispute or avenge an insult, especially in historical or samurai contexts.
彼は敵と果たし合いをすることになった。
He ended up fighting a duel with his enemy.
果たし合いの場に遅れるわけにはいかない。
I can't be late to the place of the duel.
決闘 is a more general term for a duel, often used in Western contexts, while 果たし合い carries a stronger nuance of settling a personal grudge in a traditional Japanese setting.
Compound of 果たし (masu-stem of 果たす, 'to accomplish, carry out') + 合い (continuative of 合う, 'mutually'), literally 'mutual accomplishment', referring to a decisive confrontation.