I never really thought about it, but you're right.

B&J Cherry Garcia for example, lists among other ingredients: Guar Gum, Carrageenan, and Soy Lecithin, which are thickeners and emulsifiers, and would probably meet the proposed California definition of highly processed foods

The one rival brand that I looked at, Haagen Dazs, has very simple ingredient lists. Cherry Vanilla for example has only Pectin as a recognizable thickener.