  We present a definition for second-order freeness in the quaternionic case.
We demonstrate that this definition on a second-order probability space is
asymptotically satisfied by independent symplectically invariant quaternionic
matrices.
  This definition is different from the natural definition for complex and real
second-order probability spaces, those motivated by the asymptotic behaviour of
unitarily invariant and orthogonally invariant random matrices respectively.
  Most notably, because the quaternionic trace does not have the cyclic
property of a trace over a commutative field, the asymmetries which appear in
the multi-matrix context result in an asymmetric contribution from the terms
which appear symmetrically in the complex and real cases.
