Planning for hydropower, water resources management, and climate change adaptation requires statistically unbiased hydrologic predictions. However, all hydrologic models contain systematic errors, e.g., incorrect mathematical representations of physical processes and effects of uncertainties in data sources. Statistical post-processing, or bias correction, is often used to reduce the effects of these systematic errors in model outputs. A large number of techniques for performing bias correction has been developed, primarily in the context of correcting statistical properties of independent locations. However, when bias correcting streamflow predictions within the same stream network, this assumption of spatial independence breaks down. Independently bias correcting locations from the headwaters to the mouth of a river system destroys the spatial consistency of the streamflow across a river network. We describe work toward maintaining spatial consistency in streamflow bias correction using a number of locations in the western United States. We simulate the hydrology of the Columbia River in the Pacific Northwestern United States, a river system that spans a number of hydroclimatic and flow regimes that contains a large number of flow gages. We develop a mapping from the modeled output at the gages with flow observations, which we use as the basis for training a machine learning (ML) model to perform the site-specific bias correction. We then apply the ML model to local streamflow contributions for each river segment, including river segments without flow observations. Finally, we combine the local bias corrections across the stream network, to create accumulated bias-corrected streamflow time series that are spatially-consistent across the stream network. We compare our method against several commonly used bias correction techniques to evaluate both model performance and spatial consistency.