Conscious experience seems to be single threaded, we know that brain synchronizes senses (for example sound of a bouncing ball needs to be aligned with visual of bouncing ball), but IMO it's not so obvious what is the reason for it. The point of having the experience may not be acting in the moment, but monitoring how the unconscious systems behave and adjusting (aka learning).
Haven't there been experiments on people who have had their corpus callosum severed where they seem to have dual competing conscious experiences?
Yep, folks should look up "Alien hand syndrome".
Well, serializing our experiences into memories is a big one. There's been a big project in psychology probing the boundary between conscious and subliminal experiences and while subliminal stimuli can affect our behavior in the moment all trace of them is gone after a second or two.
We have very little insight to our own cognition. We know the 'output layer' that we call the conscious self seems to be single threaded in this way, but that's like the blind man who feels the elephants trunk and announces that the elephant is like a snake.