1. Sterilize jars, and prepare lids.
2. While jars are boiling, peel carrots and radishes, and cut into 1/8-inch thick julienne strips using a mandoline or knife.
3. Bring 2 cups water, vinegar, sugar, salt, ginger, and garlic to a boil in a 6-qt. stainless steel or enameled Dutch oven.
4. Place 1 Tbsp. coriander seeds, 1 to 2 chiles, 2 garlic cloves, and one fourth of the ginger slices in each hot jar. Pack carrots and radishes tightly in jars, leaving 1/2-inch headspace. Cover vegetables with hot pickling liquid, leaving 1/2-inch headspace. Seal and process jars, processing 5 minutes.
5. Remove jars from water, and let stand, undisturbed, at room temperature 24 hours. To check seals, remove the bands, and press down on the center of each lid. If the lid doesn’t move, the jar is sealed. If the lid depresses and pops up again, the jar is not sealed. Store properly sealed jars in a cool, dark place up to 1 year. Refrigerate after opening.