Use a vegetable peeler to peel off the lime zest in long strips, avoiding the white pith beneath. Put the zest in a medium saucepan with the lime juice. Halve the mandarins and squeeze the juice into the saucepan, then place the squeezed halves into the pan as well.
Add the sugar and water and bring to a boil. Remove from heat, cover, and let steep for 30 minutes. Strain out and discard the solids.
To make a soda, put 2 to 3 tablespoons of the syrup into a large glass full of ice and top with 1 cup of seltzer, stirring well.