A white piece of paper on my table does not look as white in the evening when the light bulb is illuminating it compared to daytime when the Sun is illuminating it. The difference is much smaller than what you would get if you assume that the brain does not perform any white balance correction at all. However, it seems to me that the brain does not perfectly compensate for the change in lighting. If I set the white balance "correctly" in my camera to make the white piece of paper white, then the picture I get is not precisely the way things really look in the room.
The question is then if there is a practical fine tuning procedure to get the picture to match what we actually observe, similar to how you would go about getting a correctly exposed picture (in that case the camera measures the amount of light to determine the exposure but that would make bright objects to become a bit too dark or very dark objects would become too bright, so we have to use exposure compensation to get it right).