I don't have hard data at hand, but I think most investigations into this phenomenon show the same pattern. People identify as independent when given the choice

I don't have hard data at hand, but I think most investigations into this phenomenon show the same pattern. People identify as independent when given the choice between that, R, or D, but in practice many of those vote almost exclusively for one party. My recollection is that there are somewhat more people who identify as Independent who vote primarily for Republicans, but that might not be true.