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We present simultaneous Chandra X-ray Observatory and Hubble Space Telescope observations of three certain (X5, X7, W37) and two likely (X4, W17) quiescent neutron-star low-mass X-ray binaries (qLMXBs) in the globular cluster 47 Tuc. We study these systems in the X-ray, optical and near-ultraviolet (NUV) using the simultaneous data and additional non-contemporaneous HST data. We have discovered a blue and variable NUV counterpart to W17. We have not securely identified the eclipsing qLMXB W37 in the optical or NUV. Deeper high-resolution imaging is needed to further investigate the faint NUV excess near the centre of the W37 error circle. We suggest that a previously identified optical astrometric match to X7 is likely the true counterpart. The Halpha emission and the location of the counterpart in the colour-magnitude diagram, indicate that the secondary is probably a non-degenerate, H-rich star. This is consistent with previous results from fitting X7's X-ray spectrum. In X4, the simultaneous X-ray and optical behaviour supports the earlier suggestion that the X-ray variability is driven by changes in accretion rate. The X-ray eclipses in X5 coincide with minima in the optical/NUV light curves. Comparison of the 47 Tuc qLMXBs with the cataclysmic variables (CVs) in the cluster confirms that overall the qLMXBs have larger X-ray-to-optical flux ratios. Based on their optical/NUV colors, we conclude that the accretion disks in the qLMXBs are less prominent than in CVs. This makes the ratio of X-ray flux to excess blue optical flux a powerful discriminator between CVs and qLMXBs., Comment: accepted by MNRAS