1. Sterilize jars, and prepare lids.
2. While jars are boiling, scrub turnips thoroughly, and remove any blemishes; trim off green tops. Cut turnips into 1-inch wedges. Bring vinegar, salt, sugar, and 1/2 cup water to a boil in a 3-qt. stainless steel saucepan over medium heat, stirring to dissolve sugar and salt.
3. Place 2 shiso leaves, 1 serrano pepper, and 1 garlic clove in each hot jar. Pack turnips tightly in jars, leaving 1/2-inch headspace. Cover turnips with hot pickling liquid, leaving 1/2-inch headspace.
4. Seal and process jars, processing 15 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.