To be conscious of something, one may need some sort of processing distinct from non-conscious processing. So there could be some specialized centers for conscious processing. (Might not be localized to one spot in the brain, though.)
There are also "afferent copies" of neural activity in many regions--signals not really used for normal processing, but broadcast nonetheless for various reasons (e.g. afferent copies of motor commands seem to function as a predictive signal for where limbs will be).
Some afferent copies of neural activity would then presumably go to centers for conscious processing.
Afferent copies of the computations underlying conscious processing could be detected by centers for conscious processing, giving a conscious sense that, "Yes, I am conscious right now".
This is probably not how it works, but there is no known reason why it couldn't be, so it satisfies "what could possibly be a correlate".