Umeda et al. (Jpn J Appl Math 1:435–457, 1984 ) considered a rather general class of symmetric hyperbolic–parabolic systems: A^0z_t+∑_j=1^nA^jz_x_j+Lz=∑_j,k=1^nB^jkz_x_jx_k and showed optimal decay rates with certain dissipative assumptions. In their results, the dissipation matrices L and B^jk(j,k=1,…,n) are both assumed to be real symmetric. So far there are no general results in case that L and B^jk are not necessarily symmetric, which is left open now. In this paper, we investigate compressible Navier–Stokes–Maxwell (N–S–M) equations arising in plasmas physics, which is a concrete example of hyperbolic–parabolic composite systems with non-symmetric dissipation. It is observed that the Cauchy problem for N–S–M equations admits the dissipative mechanism of regularity-loss type. Consequently, extra higher regularity is usually needed to obtain the optimal decay rate of L^1(ℝ^3) - L^2(ℝ^3) type, in comparison with that for the global-in-time existence of smooth solutions. In this paper, we obtain the minimal decay regularity of global smooth solutions to N–S–M equations, with aid of L^p(ℝ^n) - L^q(ℝ^n) - L^r(ℝ^n) estimates. It is worth noting that the relation between decay derivative orders and the regularity index of initial data is firstly found in the optimal decay estimates.