The global ocean absorbs a significant portion of anthropogenic carbon dioxide (CO₂) emissions. Tracking the fate of absorbed CO₂ and its impacts requires an internally consistent global observational dataset spanning decades. In the Global Ocean Data Analysis Project (GLODAP), data from disparate research cruises are compared in a secondary quality-control process, adjusted for consistency where necessary and compiled into a data product. Differences between cruises are quantified with a crossover analysis, comparing data at depth (where natural variability is minimal) from nearby sampling stations, and an inversion algorithm calculates a set of adjustments that would minimise these differences globally. The adjustments are reviewed by an expert committee and a subset applied to produce the final data product. The previous major version (GLODAPv2) used a weighted least squares (WLSQ) approach for the inversion. However, several issues became apparent when applied to the GLODAPv3 dataset, primarily significant regional biases in calculated adjustments. To address these issues, a new inversion algorithm called furthest-first (FF) has been developed for use in GLODAPv3, which has been implemented in a freely available, open-source Python package (xover). Here, we describe the FF approach and test it on simulated datasets and by comparing it to WLSQ, finding that it produces accurate adjustments. We also show how the FF approach can be adapted (i) to avoid the regional biases that appear in WLSQ, (ii) to find an optimal set of adjustments accounting for the fact that some cruises will ultimately not be adjusted, and (iii) to approximately preserve selected trends in the cruise-by-cruise differences. Points (i) and (ii) above are addressed by a two-step variant of the approach denoted FF₂, which was applied in GLODAPv3.  While FF represents a marked improvement on WLSQ for the GLODAP application, no algorithm can completely replace the role of expert judgement in the inversion process, for example when selecting convergence criteria, which cruises are permitted to be adjusted, and which trends should be preserved.