Put apple cores, cardamom pods, tea leaves, and water in a small saucepan. Bring to a boil over high heat, then continue to boil, uncovered, until the liquid reduces by about half, about 20 minutes.
Strain through a fine mesh strainer into a medium bowl, pressing on the softened apple cores with a wooden spoon to release additional liquid.
Pour 1 cup of apple liquid into the saucepan. Add sugar and return the pan to the stove over medium-high heat. Bring to a simmer and cook, stirring, until sugar dissolves and the liquid gets slightly thicker but still quite pourable, about 5 minutes. Cool and strain into a bottle for storing. Keeps in refrigerator for 2-3 weeks.
