Let E and E′ be elliptic curves over Q with complex multiplication by the ring of integers of an imaginary quadratic field K and let Y = Kum(E×E′) be the minimal desingularisation of the quotient of E×E′ by the action of −1. We study the Brauer groups of such surfaces Y and use them to furnish new examples of transcendental Brauer–Manin obstructions to weak approximation. This is joint work with Mohamed Alaa Tawfik.