The steady inspiratory flow through an anatomically accurate model of the human airways was studied experimentally at a regime relevant to deep inspiration for aerosol drug delivery. Magnetic resonance velocimetry was used to obtain the three-component, mean velocity field. A strong, single-sided streamwise swirl was found in the trachea and persists up to the first bifurcation. There, the swirl and the asymmetric anatomy impact both the streamwise momentum distribution and the secondary flows in the main bronchi, with large differences compared to what is found in idealized branching tubes. In further generations, the streamwise velocity never recovers a symmetric profile and the relative intensity of the secondary flows remains strong. Overall, the results suggest that, in real human airways, both streamwise dispersion (due to streamwise gradients) and lateral dispersion (due to secondary flows) are very effective transport mechanisms. Neglecting the extrathoracic airways and idealizing the bronchial tree may lead to qualitatively different conclusions.