  In six-dimensional F-theory/heterotic string theory, half-hypermultiplets
arise only when they correspond to particular quaternionic K\"ahler symmetric
spaces, which are mostly associated with the Freudenthal-Tits magic square.
Motivated by the intriguing singularity structure previously found in such
F-theory models with a gauge group $SU(6)$,$SO(12)$ or $E_7$, we investigate,
as the final magical example, an F-theory on an elliptic fibration over a
Hirzebruch surface of the non-split $I_6$ type, in which the unbroken gauge
symmetry is supposed to be $Sp(3)$. We find significant qualitative differences
between the previous F-theory models associated with the magic square and the
present case. We argue that the relevant half-hypermultiplets arise at the
$E_6$ points, where half-hypermultiplets ${\bf 20}$ of $SU(6)$ would have
appeared in the split model. We also consider the problem on the non-local
matter generation near the $D_6$ point. After stating what the problem is, we
explain why this is so by using the recent result that a split/non-split
transition can be regarded as a conifold transition.
