Detect and remove overlaps resulting from polygons with crossing edges
For context: polygons are meant to be in a print product (offset print). In order to avoid misregistration artefacts (white flashes when different colored print plates are not perfectly aligned) we set the edges of our polygons to overprint (this means different colors are not knocked out but printed on top of each other. This ensures that if one printing plate would shift a bit we'd still have a portion of the differently coloured neighbour polygons edge showing up (this works for shifts up to and equal to a shift of half of the contour strength of the edges).
For this concept of overprinting to be working we need to ensure that all the polygons are contiguous, meaning that the edges of neighbouring polygons always evolve next to each other and don't cross edges (aka producing overlaps) as without that constraint white flashes from misregistration of the printing plates can occur again where only edges of one polygon of a neighbouring pair is present.
Running the Remove Overlap (multiple) (Analysis) tool with the Thiessen method (or any of the alternative methods) will result in 2 valid features (below), while 4 have no geometry anymore, as the tool combines features that are on top of each other. This tool produces the target result for overlapping features with crossing edges that are not fully contained within another feature in the layer.
But we want to remove overlaps only where edge-crossing is happening. Fully in another polygon "enclosed" polygons are allowed. So the overlaps of 2&3 and 1&5 would need to be resolved.
In my post @BrennanSmith1 proposed a well working ArcPy solution. It would be great to implement this one into a tool of the GUI, so a more wider audience could simply use it.