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I have two layers that look like this:

before1

I want the intersection (or whatever the correct term may be) to look like this where the output polygons are the same as the original base layer, just with added attributes if there is any kind of intersection. Preferably, there'd be a way to say something like, if there is over 50% coverage, take the property, if not, no property.

after1

Similarly, this area has a lot of intersections where it shouldn't. (The C4 applies to the dark shapes and the numbers apply to the lighter shapes. The darker wide line is where the two number polygons overlap.)

before2

Ideally, the "fuzzy" intersection would work similarly here where the polygon below gets the added property of the polygon above that overlaps the most with it. Again, the output should be with the same polygon shapes as the base layer.

after2

How can I achieve these results? I've looked into Union, Intersect, Clip, Identity and such, but none seem to be right. Perhaps there is an option in one of these that I am missing? I can use either ArcGIS or QGIS.

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Using the tools in ArcGIS, if I understand your question correctly: You can use 'Tabulate Intersection' to calculate the % of area overlap between your two polygon layers (using FID as your zone field) and join the output to the appropriate polygon layer. Then use spatial join to join the attributes from one layer to the other. You can then delete any records under your % cutoff.

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