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I am trying to find a method (in QGIS and/or GRASS) to join two layers of polygons into a resulting layer with merged overlapping polygons. It's hard to explain in words, so I've drawn a simplified graphic

enter image description here

Has anyone come up with a working solution for this need?

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You can do such operations with v.overlay. In the online manual, see the example "EXAMPLE POLYGON TO POLYGON UNION. Polygon union of urban area and Census 2000 areas (North Carolina dataset)" which is similar to your request.

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