We study how the structure and variability of magnetohydrodynamic (MHD) turbulence in accretion discs converge with domain size. Our results are based on a series of vertically stratified local simulations, computed using the athena MHD code, that have fixed spatial resolution, but varying radial and azimuthal extent (from ΔR= 0.5H to 16H, where H is the vertical scale height). We show that elementary local diagnostics of the turbulence, including the Shakura–Sunyaev α parameter, the ratio of Maxwell stress to magnetic energy and the ratio of magnetic to fluid stresses, converge to within the precision of our measurements for spatial domains of radial size Lx≥ 2H. We obtain α≃ 0.02–0.03, consistent with other recent determinations. Very small domains (Lx= 0.5H) return anomalous results, independent of spatial resolution. This convergence with domain size, however, is only valid for a limited set of diagnostics: larger spatial domains admit the emergence of dynamically important mesoscale structures. In our largest simulations, the Maxwell stress shows a significant large-scale non-local component, while the density develops long-lived axisymmetric perturbations (‘zonal flows’) at the 20 per cent level. Most strikingly, the variability of the disc in fixed-sized patches decreases strongly as the simulation volume increases, while variability in the magnetically dominated corona remains constant. Comparing our largest local simulations to global simulations with comparable spatial resolution, we find generally good agreement. There is no direct evidence that the presence of curvature terms or radial gradients in global calculations materially affect the turbulence, except to perhaps introduce an outer radial scale for mesoscale structures. The demonstrated importance of mean magnetic fields – seen in both large local and global simulations – implies, however, that the growth and saturation of these fields is likely of critical importance for the evolution of accretion discs.