Safe to say the dirt is clay. What do you plan to do with the dirt? Sod? Shrubs? What caused the compaction?

Humic shale ore will help loosen the clay, but that will eventually wash out and you'll be back to where you started. IIRC, the people from Wisconsin were saying that the pH of their ground was alkaline, so I wonder if gypsum is the way to go.

Organic amendments will also help denature the clay, but if the area is getting traffic, you'll need a better long term solution. You might want to find sources of silt and sand and modify the clay until you have a loam mixture. I don't think loam will compact the way that clay does.

The property behind us is at a higher elevation. This area where the properties abutt eachother has a slope where our fence sits. The slope caused erosion of neighbors soil onto our fence. So, our solution, on our side of the fence anyway, was a small bumped out retaining wall with a planting bed atop, which was backfilled with mostly the surrounding topsoil in that area. So, basically, the areas at the base of the retaining wall is not topsoil, but a layer that sat maybe a foot under the original sloping grade. We do have somewhat clay soil, but of course, this area is much more compacted than the rest of my yard, for reasons stated.

I don't know what "IIRC" means.

I had a bag of gypsum laying around so that's what I used, but I realize that's only a long term solution which I'll maybe start seeing benefits in two more years.

Area doesn't get much traffic. I planted there when it was well watered, but some areas especially in hot sun are really compacted, and I admit I didn't do a lot of watering this past year. By modifying the clay, I'm assuming you are referring to tilling it up and adding silt and sand. I don't want to dig everything up, some plants are already established, and so far seem to be doing fairly well in the existing soil, so I was hoping an easier solution was to just add organic matter on a regular basis??? Sounds like this is also not a quick fix, but I can work with it.

IIRC - if I remember correctly

Kim, this my not be the best solution, but it's what I would do (if I understand your situation correctly). First, don't work your soil, or walk on it, when it's wet. This will cause compaction for sure. If your beds are too wide to work from the sides you could put some stepping stones throughout so you don't step on the soil. When the garden is reasonably dry, but not bone dry, I would use my garden fork or my hay fork depending how compacted the soil is. I would push the fork into the soil as far as it will go and wiggle it around a bit. Put down a couple of inches of good compost and go over it again with the fork wiggling it to work the compost into the soil a bit. I might also dig a hole here and there between the plants and put some veggie scraps and used coffee grounds there and cover it with soil. I think of this as feeding the worms. If you encourage earthworms they will help with loosening the soil.

There is no magic way to build soil, just keep adding compost and encourage the worms. Try not to use chemicals too much as the worms don't like them! :-)

Gypsum is only valuable when dealing with sodic (high saline) soils - otherwise it is waste of time and effort. Adding sand to heavy soils only works if the quantity is significant (most sources suggest a 50-50 mix) and the sand of large particle size.

luckygal's suggestion is about as good as you will get. Improving soil tilth, especially with compacted or heavy soils, is not an overnight process. Keep applying the compost and continue to plant - plant roots will go along way in breaking up and loosening the soil as well.

Here is a link that might be useful: the myth of gypsum

Gypsum does nothing to loosen the clay soils of Wisconsin, although if you had some laying around it might be of some other benefit in the soil rather than in a shed. What you soil needs is organic matter, lots of organic matter, adn your idea of laying some composted manure on top can help, but that is a mulch. Mulch is any material that is laid on the soils surface and some mulches are mroe beneficial to soil than others.
A good soil will have between 5 and 8 percent organic matter in it, so work toward that goal and that will make your clay soil into something workable, loose, and friable.

There are a few plants that are supposed to be pretty good at breaking up compacted clay soil. Borage is a lovely flowering plant that has a long tap root that gets in and breaks up the soil. The bees love it too. If you plant some there and leave it, it will self seed for next year too, if you like. You can also cut and compost the plants. They are big, with lots of greens for your pile.

Another thing to try would be broad beans/fava beans. They are supposed to be great for breaking up clay soil, and they are legumes and fix lots of nitrogen to boot. I have never grown them, but I understand that they are pretty easy to grow, and actually prefer the cool weather. You should be able to plant them around the same time as spring peas, from what I have read.

Finally, it is common to read that potatoes are known to be good at breaking up clay soil.