The vertebrate adaptive immune system modifies the genome of individual B cells to encode antibodies binding particular antigens. In most mammals, antibodies are composed of a heavy and a light chain which are sequentially generated by recombination of V, D (for heavy chains), J, and C gene segments. Each chain contains three complementarity-determining regions (CDR1-3), contributing to antigen specificity. Certain heavy and light chains are preferred for particular antigens. We considered pairs of B cells sharing the same heavy chain V gene and CDRH3 amino acid sequence and isolated from different donors, also known as public clonotypes. We show that for naive antibodies (not yet adapted to antigens), the probability that they use the same light chain V gene is ~10%, whereas for memory (functional) antibodies it is ~80%. This property of functional antibodies is a phenomenon we call light chain coherence. We also observe it when similar heavy chains recur within a donor. Thus, though naive antibodies appear to recur by chance, the recurrence of functional antibodies reveals surprising constraint and determinism in the processes of V(D)J recombination and immune selection. For most functional antibodies, the heavy chain determines the light chain.