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.