  We show that a pair of almost commuting self-adjoint, symmetric matrices is
close to a pair of commuting self-adjoint, symmetric matrices (in a uniform
way). Moreover we prove that the same holds with self-dual in place of
symmetric. The notion of self-dual Hermitian matrices is important in physics
when studying fermionic systems that have time reversal symmetry. Since a
symmetric, self-adjoint matrix is real, we get a real version of Huaxin Lin's
famous theorem on almost commuting matrices. Similarly the self-dual case gives
a version for matrices over the quaternions.
  We prove analogous results for element of real C^*-algebras of "low rank." In
particular, these stronger results apply to paths of almost commuting Hermitian
matrices that are real or self-dual. Along the way we develop a theory of
semiprojectivity for real C^*-algebras.
