We consider the effect of a short antiferromagnetic correlation length xi on the electronic band structure of the underdoped cuprates. Starting with a Fermi-surface topology consistent with magnetic-quantum-oscillation data, we show that a reduced xi gives an asymmetric broadening of the quasiparticle dispersion, resulting in simulated ARPES data very similar to those observed in experiment. Predicted features include "Fermi arcs" close to ak=(pi/2,pi/2), where a is the in-plane lattice parameter, without the need to invoke a d-wave pseudogap order parameter. The statistical variation in the k-space areas of the reconstructed Fermi-surface pockets causes the quantum oscillations to be strongly damped, even in very strong magnetic fields, in agreement with experiment.