noun, noun which may take the genitive case particle 'no'
unrelated by blood; step-relationship
Refers to a relationship where the people involved are not biologically related, often used in the context of step-families or adoptive relationships. The phrase is somewhat literary and fixed.
彼女はなさぬ仲の子供たちを実の子のように育てた。
She raised the children, who were not her own by blood, as if they were her real children.
なさぬ仲だからこそ、気を遣うこともある。
Precisely because they are not blood-related, there are times when one must be considerate.
血の繋がり means 'blood relation' and is a neutral, common term. なさぬ仲 specifically emphasizes the absence of such a tie, often with a nuance of a formed relationship despite that.
義理の is a prefix for 'in-law' or 'step-' relationships (e.g., 義理の母). なさぬ仲 is a more literary, noun-based expression describing the state of not being blood-related.
From 生さぬ (nasanu), the classical negative form of 生す (nasu, 'to give birth'), meaning 'did not give birth to', and 仲 (naka, 'relationship'). Literally 'a relationship where one did not give birth'.