We present an analysis of high-quality photometry for globular clusters (GCs) in the Virgo cluster core region, based on data from the Next Generation Virgo Cluster Survey (NGVS) pilot field, and in the Milky Way (MW), based on Very Large Telescope/X-Shooter spectrophotometry. We find significant discrepancies in color-color diagrams between sub-samples from different environments, confirming that the environment has a strong influence on the integrated colors of GCs. GC color distributions along a single color are not sufficient to capture the differences we observe in color-color space. While the average photometric colors become bluer with increasing radial distance to the cD galaxy M87, we also find a relation between the environment an...