Peel the prickly pears: Start by slicing off both ends, then slit the skin down the entire length. You should be able to simply unroll the skin off of the fruit from that point - it should come loose quite easily.
Place the peeled prickly pears into a large pot, and cover with water by about an inch (they will float, so you will need to hold one down to check the water depth).
Bring to a boil over medium-high heat. Reduce heat to low, and simmer for 15 minutes. While the fruit is simmering, attempt to break up the larger pieces as they begin to soften (a large spoon or a potato masher works well for this)
Remove from heat, and pour through a fine sieve into a large pitcher or similar container. Use the back of a spoon to squeeze as much liquid from the pulp through the sieve and into the pitcher as possible. Discard pulp and seeds when you are finished.
Add as much cold water as needed to bring the total volume up to about 2 quarts. Sweeten to taste (we sweeten ours rather lightly, usually starting with about 1/4 cup of sugar and gradually increasing as desired from there).
Refrigerate until chilled.