The treatment of the fit residual is a central problem in DOAS retrievals from hyperspectral satellite instruments. At the precision of instruments such as TROPOMI, the residual of a classical fit is no longer an approximately white background, but splits into systematic structures of physical and instrumental origin and a correlated random noise. Classical DOAS treats both in the same way, which limits its performance for weak absorbers retrieved under difficult conditions. This study presents XDOAS, a complete retrieval that unifies two recent approaches to this problem, covariance weighting and principal component analysis (PCA), within a single linear formulation. The PCA components fit the systematic structures out of the residual explicitly, while a regularised covariance down-weights the remaining correlated noise. Classical DOAS, covariance-weighted DOAS, and PCA-DOAS emerge as limiting cases of XDOAS. The algorithm is applied to the retrieval of OClO slant column densities from TROPOMI/S5P, a weak absorber measured at large solar zenith angles where the residual structures are most pronounced. Evaluated against the three reference retrievals on TROPOMI measurements using identical fit parameters, XDOAS reaches the precision of the best advanced retrievals while being the only one that also gives an uncertainty consistent with the spectral fit quality of each pixel. A consistency test between the RMS residual and the retrieval uncertainty separates the four retrievals, covariance-weighted DOAS gives uncertainties that are decoupled from the residual (<em>R</em>² = 0.002), PCA-DOAS gives uncertainties that are fixed to the residual because the PCA components fit out part of the random noise (<em>R</em>² = 0.991), and XDOAS is the only retrieval whose uncertainty follows the spectral fit quality of each pixel with a moderate pixel-to-pixel spread (<em>R</em>² = 0.977). The comparison against the S5P-PAL OClO product based on empirical post-processed DOAS retrieval, shows close agreement (<em>R</em>² = 0.922). Where the OClO signal is strong, over the polar vortex, the two products agree almost exactly (slope 1.02, <em>R</em>² = 0.946), while XDOAS gives a lower RMS residual and a smoother uncertainty field specially, most of all under the low-illumination conditions in which OClO is measured. XDOAS is the baseline algorithm for the operational OClO product of the Copernicus Sentinel-5 mission from the EUMETSAT AC-SAF. XDOAS is a general framework that can be applied to any trace gas with narrow absorption bands, and in particular to measurements under extrema viewing conditions typically in geostationary missions like Sentinel-4, GEMS, and TEMPO.