、。
Even if our legal family ties have been severed, the bonds of past days remain intact.
Literal
Family-register [on-上] [attributive-の] tie [even-though-ても] has-been-broken, former-days [attributive-の] bonds [topic-は] as-they-are.
A rueful statement about a broken legal relationship. 戸籍 ('family register') is the Japanese civil registry system recording births, marriages, and legal family relationships. 縁 ('tie, connection, bond, fate') has both practical and fateful connotations. ~上の ('on the X level, in terms of X') is a formal attributive — 戸籍上の縁 = 'legal family tie.' 昔日 (せきじつ) is literary for 'former days, the past.' そのまま ('as is, unchanged'). 切れる ('to be severed').