I get asked this question quite a bit. My answer is always "YES!".
What I mean is this: the colors you see on my final image are real. Theyare the colors I saw in my mind's eye when I composed the original image.
This answer is somewhat unsatisfying to people, but it's the only one I can honestly give.
First, we have to realize that the human eye perceives a tiny portion of the full radio wave spectrum. Google "light spectrum" and you'll find hundreds of images like the one below.
All sorts of other species of fauna see different parts of the spectrum. Bees and butterflies, for example, can see into the ultraviolet spectrum. Snakes can "see" infrared, though it happens through sensors in the skin.
Most importantly, "colors" are no more than conventions for describing particular sets of wave lengths. Not only that, but the color you see might be light that is reflected off an object, OR it might be light emitted from that object. In other words, a visible color might be an intrinsic part of the object, or it might be extrinsic to the object.
Thus it is very difficult to determine any objective, "real" basis for color.
Now add to this the photographic process. There are many purists who try to make the argument that photography must somehow reflect the real world exactly. While this is obviously an important issue for photo journalists, it is irrelevant to those who produce "art" photography, i.e., ALL types of photography other than documentary.
Even in the old days of film, artists like Ansel Adams viewed the negative as only part of the process. In fact, he wrote an entire book on the editing negatives, and another one on printing him. His emphasis in these books is not the absolutely realistic rendition of the world as it is, but rather the artistic creation of the artist's vision when the photograph was created. Do you think "Moonrise Over Hernandez" looked that way straight out of the camera? Think again.
The process becomes even more involved in the digital age. If you take a photo with your smart phone or any point-and-shoot camera, your image has already been considerably modified by the software in the device. If you don't believe it, compare a raw image from a higher end DSLR with a jpeg from your smart phone. The latter has much more saturation, higher contrast, perhaps some sharpening, etc.
I shoot everything in raw mode. The raw images are not much to look at. They contain all the color information, but the colors and contrast are quite muted. Take a look at the original raw version of "Bullet Train":
And now compare it to the processed version:
Which has the "real" colors? If you mean by that, which one most closely resembles the light reflecting from the surface of the object you saw when you took the photograph, I honestly couldn't answer. I suppose it's somewhere in between the two versions.
But if you mean what are the "real" colors to me as a photographer, then the answer is clear. It's the ones you see in the final version. And anyone who's something other than a documentarian, if they're being honest, would say the same thing.