We show that the distance in total variation between (Tr U, √12 Tr U2, . . ., √m Tr Um) and a real Gaussian vector, where 1 U is a Haar distributed orthogonal or symplectic matrix of size 2n or 2n + 1, is bounded by 「 (2 mn + 1)− 12 times a correction. The correction term is explicit and holds for all n ≥ m4, for m sufficiently large. For n ≥ m3 we obtain the bound (mn)−c √ mn with an explicit constant c. Our method of proof is based on an identity of Toeplitz + Hankel determinants due to Basor and Ehrhardt, see (Oper. Matrices 3 (2009) 167–86), which is also used to compute the joint moments of the traces.