In this paper we derive a general limit characterization of principal solutions at infinity of linear Hamiltonian systems under no controllability assumption. The main result is formulated in terms of a limit involving antiprincipal solutions at infinity of the system. The novelty lies in the fact that the principal and antiprincipal solutions at infinity may belong to two different genera of conjoined bases, i.e., the eventual image of their first components is not required to be the same as in the known literature. For this purpose we extend the theory of genera of conjoined bases, which was recently initiated by the authors. We show that the orthogonal projector representing each genus of conjoined bases satisfies a symmetric Riccati matrix differential equation. This result then leads to an exact description of the structure of the set of all genera, in particular it forms a complete lattice. We also provide several examples, which illustrate our new theory.