Some of the foils in All the King's Men are directly announced as foils. Like this: "Tiny Duffy became, in a crazy kind of way, the other self of Willie Stark, and all the contempt and insult which Willie Stark was to heap on Tiny Duffy was nothing but what one self of Willie Stark did to the other self" (2.492).
Jack isn't literally suggesting that Tiny is only an illusion, or an aspect of Willie's personality. He's trying to provide an explanation as to why Willie keeps company with this questionable man. Jack claims that in Willie's mind Tiny is his foil. Tiny represents everything Willie wants not to be.
Jack and Tiny are foils in All the King's Men. This connection, though, is not clear until the end of the novel when Tiny has become governor, and wants Jack to be for him what he was for Willie. Jack says he knows that Tiny knows "the nightmare truth, which was that we were twins bound together […] intimately and disastrously […]" (10.197). This marks a turning point for Jack. He was willing to lay all the blame for Willie's death on Tiny. Having someone to accuse and look down on makes him feel good. At this moment he realizes that he has just as big a roll in Willie's death as Tiny: their shared knowledge of the facts that lead to Willie's death twins them indefinitely. So long as Jack condemns Tiny, he must also condemn himself.