The essence of an idependent mind lies not it what it thinks, but how it thinks.
Christopher Hitchens
I would however argue that a independent mind committed to dealing in reality will tend to substantively agree with other similarly dedicated minds.
I'm sure that somewhere, someplace, is a person who considers themselves an independent thinker believes some conspiracy theory about disguised lizard aliens among us, but I'd have some difficulty respecting their beliefs.
There are independent minds out there that believe in homeopathy, crystals, cupping and all sorts of silliness. Independence does not require discerning.