Mix together the pomegranate juice and sugar in a small pot over the stove, and stir over low to medium heat until the sugar has completely dissolved.
Optionally thicken the mixture more by bringing the heat down to low and simmering the mixture until it has thickened to your liking. Keep in mind that the mixture will thicken further when cooled.
Add lemon or lime juice, to taste, towards the end of the cooking process.
Pour the mixture into bottles and store in the refrigerator for up to a month or more.
To make your own pomegranate juice from scratch, cut open fresh pomegranates and remove the seeds. Blend the seeds with a blender and strain out the juice through a cloth or mesh strainer. Homemade grenadine can be used in cocktails like the Tequila Sunrise or mocktails like the Shirley Temple. You can extend the shelf life further by adding some vodka to the mixture. You'll end up with more of a pomegranate liqueur which can keep almost indefinitely if you add enough vodka.