Put all pickling-liquid ingredients plus 2 1/3 cups water into a large pot and bring to a boil, covered. Reduce heat and simmer, stirring occasionally, until salt and sugar is dissolved. Let cool and settle.
Line a colander with cheesecloth, set it over a large bowl, and slowly pour in liquid, leaving most of turmeric sediment behind in pot.
Mix vegetables in a large nonreactive bowl, then divide between 2 qt-size wide-mouth mason jars. Bring pickling liquid to a simmer and pour over the vegetables, again leaving any turmeric sediment behind. Chill 24 hours before using. Serve in a bowl, with some of the liquid included.
Make ahead: Up to 1 month, chilled.