We have observed very low disorder in high quality quantum well tubes (QWT) in GaAs-Al0.4Ga0.6As core-multishell nanowires. Room-temperature photoluminescence spectra were measured from 150 single nanowires enabling a full statistical analysis of both intra- and inter-nanowire disorder. By modelling individual nanowire spectra, we assigned a quantum well tube thickness, a core disorder parameter and a QWT disorder parameter to each nanowire. A strong correlation was observed between disorder in the GaAs cores and disorder in the GaAs QWTs, which indicates that variations in core morphology effectively propagate to the shell layers. This highlights the importance of high quality core growth prior to shell deposition. Furthermore, variations in QWT thicknesses for different facet directions was found to be a likely cause of intra-wire disorder, highlighting the need for accurate shell growth.