Almost commuting orthogonal matrices

We show that two almost commuting real orthogonal matrices are uniformly close to exactly commuting real orthogonal matrices. We prove the same for symplectic unitary matrices. This is in contrast to the general complex case, where not all pairs of almost commuting unitaries are close to commuting pairs. Our techniques also yield results about almost normal matrices over the reals and the quaternions. We conclude with an example where the K-theoretical obstructions to approximation cannot be avoided. Our example is inspired by the physical systems known as topological superconductors.