The soil in my backyard has very few rocks but the clay is so hard and dense it may as well be a brick wall.

When it's wet, but not saturated - like 1-2 days after a rain - you can decompact the soil with a strong metal broadfork and leave the soil in large block aggregates. This keeps the soil structure and maintains some fungal web connections. Add nutrients, wood chips, stick and sand below aggregates and in cracks. Cover with compost and plant clover to cover.

This is amazing info thank you.

What is the purpose of planting clover?

Clover is a nitrogen fixing plant - used to be that you’d plant clover for a year in between other crops to make your soil fertile again.

(It’s the bacteria in the roots that do the actual nitrogen chemistry.)

Really good advice above. But if you want to cheat a bit, I used https://www.thompson-morgan.com/p/vitax-clay-breaker--25-kg-... on my heavy clay garden and it helped a lot, in combination with extra organic material etc.

Clover fixes nitrogen and roots help stabilize the voids in the soil. They sell seed mixes called "ground cover mix" that includes other plants and will help keep the soil from recompacting when it rains and keeps weeds at bay.

Are you sure it is just clay? Sandy clay packs even tighter than just clay: https://www.uaex.uada.edu/farm-ranch/crops-commercial-hortic...

"If you mix sand into clay, the clay particles will fill in all the open spaces between the sand particles and often the clay will act as a ‘glue’ sticking all particles together, ultimately resulting in a more dense soil."