Models of biological coevolution have recently been proposed and studied, in which a species is defined by a genome in the form of a finite bitstring, and the interactions between species i and j are given by a fixed matrix with independent, randomly distributed elements M_ij. A consequence of the stochastic independence is that species whose genotypes differ even by a single bit may have completely different phenotypes, as defined by their interactions with the other species. This is clearly unrealistic, as closely related species should be similar in their interactions with the rest of the ecosystem. Here we therefore study a model, in which the M_ij are correlated to a controllable degree by means of a local averaging scheme. We calculate, both analytically and numerically, the correlation function for matrix elements M_ij and M_kl versus the Hamming distance between the bitstrings representing the species. The agreement between the analytical and numerical calculations is excellent for correlations of limited range, but explainable differences arise for correlation ranges that are a significant fraction of the length of the bitstring. We compare long kinetic Monte Carlo simulations of coevolution models with uncorrelated and correlated interactions, respectively. In particular, we consider the probability density for the lifetimes of individual species. The species-lifetime distribution is close to a power law with an exponent near −2 over eight decades in time in both uncorrelated and correlated cases. The durations of quasi-steady states and power spectral densities for the diversity indices display noticeable differences. However, some qualitative features, such as 1/f behaviour in power spectral densities for the diversity indices, are not affected by the correlations in the interaction matrix.