We show that a pair of almost commuting self-adjoint, symmetric matrices are
close to commuting self-adjoint, symmetric matrices (in a uniform way).
Moreover we prove that the same holds with self-dual in place of symmetric.
Since a symmetric self-adjoint matrix is real, the former gives a real version
of Huaxin Lin's famous theorem on almost commuting matrices. There are
applications to physics of Lin's original theorem and both new cases.