Cut each radish into halves or quarters. In a large stainless steel or glass bowl, combine the 2 1/2 cups water and the kosher salt. Add radishes. Cover and refrigerate for 1 to 2 hours. Drain and rinse radishes; set aside.
In a medium saucepan, combine 1 1/2 cups water, white vinegar, honey, red wine vinegar, and peppercorns. Bring to boiling, stirring occasionally. Reduce heat, cover, and simmer for 15 minutes.
Pack radishes into clean hot half-pint jars, leaving 1/4-inch headspace. Ladle hot vinegar mixture into jars, being sure to get some of the peppercorns into each jar and maintaining the 1/4-inch headspace. Discard any remaining vinegar mixture. Wipe jar rims; adjust lids.
Process filled jars in a boiling-water canner for 10 minutes (start timing when water returns to boiling). Remove jars from canner; cool on wire racks.