Turbomachinery components are inevitably subjected to various sources of manufacturing errors. The resultant variations in blade geometry eventually lead to performance degradation. This is especially true for transonic centrifugal compressor impellers where the complex interaction between the geometric variations and shock waves may amplify such degradation. However, relevant studies on uncertainty quantification or robust design optimization of centrifugal impellers are quite rare. The mechanisms of how the realistic manufacturing errors influence the impeller aerodynamic performance are not clear. In addition, most existing studies have considered a fixed level of manufacturing tolerance applied to the impeller blade surface, which neglects the combined effect of blade geometry and manufacturing tolerance on impeller performance. In this study, a collaborative robust design optimization was performed for a transonic centrifugal impeller considering realistic manufacturing errors. The realistic manufacturing error field was first modeled based on the measurements of 92 centrifugal impellers. With a combination of computational fluid dynamics simulation and the non-intrusive polynomial chaos method, the influence of manufacturing errors on impeller performance and flow field variations was quantified. To save computational costs for the uncertainty quantification, a dual dimensionality reduction method was proposed to reduce the dimensionality of uncertainties. Finally, blade angles and the manufacturing tolerance of the impeller were collaboratively optimized to enhance the impeller performance robustness against manufacturing errors. The results show that the impeller aerodynamic performance exhibited a downward trend in the presence of manufacturing errors. The flow mechanisms responsible for this trend were mainly associated with increased intensity of the shock waves near the inducer blade tip. Such shock waves were significantly alleviated by reducing impeller inlet blade angles without the loss of pressure rise capability. Consequently, the impeller performance robustness against manufacturing errors was enhanced with the standard deviation of polytropic efficiency being reduced by 35% at a lower anticipated manufacturing cost due to increased tolerance allowance.