Dig three or four holes with a post hole digger 24 inches deep along the side of the patio opposite the roof of the house. Space the holes 4 feet apart. Fill the bottom 2 inches of the holes with gravel. Tamp the gravel down with the end of a wooden board. Place a 9-foot 4-by-4-inch board in each hole. Fill the hole with a mixture of gravel and soil. Compact the soil around the posts.
Nail a wooden board the length of the distance from the first to the last post to the eaves under the roof. Nail another board across the top of the posts. Nail more boards from the posts to the board attached to the house. Space the boards running across to the top of the frame the same width as the lattice work.
Fill the gaps across the top of the frame with lattice panels and screw them down to the boards with an electric screwdriver. Screw more lattice work pieces between the posts to create a wall.
Stain or paint the wooden posts and lattice work with a waterproofing product, which will protect the wood from the weather. Remove the grass and dig up the soil along the edge of the lattice wall. Plant a row of vines and climbing plants. As the plants grow upward, train them to grow up the side of the lattice wall. Once large enough, train them to grow over the top of the structure to increase the shade.
Plant bushes along one of the open edges of the patio to increase privacy. Rose bushes and other hedge type plants work well for this purpose. Place a small fountain containing a pool with goldfish in one of the corners under the shade. Put a portable fire pit in the center of your cove. Hang wind chimes and lightweight plant pots from the lattice work roof. Move your patio furniture under the shade and enjoy.