Gently wash limes in cold water, then peel, avoiding as much of the pith as possible. (It’s helpful to cut the polar ends off first, where the stem attaches and the opposite ends.)
Cut limes in half and juice them. This should yield around 6 cups of juice.
In a non-reactive, coverable container, add sugar to juice and stir until fully dissolved, 6 to 10 minutes.
Crush peels up in handfuls to release the oils as you add them into the juice mix. Stir a bit to initiate extraction of the oils.
Wash the ginger, then shred it in a blender or food processor (no need to peel), employing some of the lime juice to allow it to liquefy, and add it into the lime mixture, stirring well.
Cover and refrigerate for 24 hours.
When ready, strain the cordial off from the peels through a fine mesh strainer or chinois.
Funnel cordial into covered container or cappable bottles and return to the refrigerator for another day to cure, before using.