  A \emph{Latin square} is a matrix of symbols such that each symbol occurs
exactly once in each row and column. A Latin square $L$ is
\emph{row-Hamiltonian} if the permutation induced by each pair of distinct rows
of $L$ is a full cycle permutation. Row-Hamiltonian Latin squares are
equivalent to perfect $1$-factorisations of complete bipartite graphs. For the
first time, we exhibit a family of Latin squares that are row-Hamiltonian and
also achieve precisely one of the related properties of being
column-Hamiltonian or symbol-Hamiltonian. This family allows us to construct
non-trivial, anti-associative, isotopically $L$-closed loop varieties, solving
an open problem posed by Falconer in 1970.
