The "rule" that real images are always inverted is not correct. That rule might work when you have only a single optical element (like a lens), but not necessarily when you have two or more.
Take a look at this:
![enter image description here](https://i.stack.imgur.com/9OnDe.png)
That's a real, upright (aka erect) image labeled $I_B$. You can tell it's real because the rays at the final image actually converge at that physical location, unlike virtual images whose location of "convergence" does not actually have physical rays passing through it (only the "backtracking" rays one typically draws).
As for your projector and mirror, you can draw a ray diagram carefully (if you know the internal workings of a projector) and apply the same test to see if the image is virtual or real. But do remember that projectors can be modified so that the image you see is inverted to accommodate mirrors and such. Anyway, to get something projected on to a screen, I do believe the image needs to be real, so I would say it is indeed a real image you're seeing.
Finally, here's a simple example of two elements that again goes against the rules in your book:
![enter image description here](https://i.stack.imgur.com/0ySUT.gif)
The situation above has a real, inverted image using a lens and mirror (a mirror!). Thus, the rules you've been given aren't general enough to deal with multiple-element situations.
Note the focal length of the spoon will be only a few centimetres if that.
If you can get close enough so that your face is closer than the focal length of the spoon then you would see a magnified and upright image which is virtual.
This is the arrangement used for a shaving or make up mirror but they have larger focal lengths.
![enter image description here](https://i.stack.imgur.com/7hIh4.jpg)
However you are more likely to be seeing a diminished and inverted image which is real and located between your face and the spoon.
Your face will be greater than two focal length from the spoon.
![enter image description here](https://i.stack.imgur.com/jsfuC.jpg)
The real image appears to be in the vicinity of the spoon but you can located by using the no-parallax method as described here for a concave mirror and for a convex lens here which can be adapted for the spoon.
Such attempts at the location of the real image will be more difficult because of the aberrations caused by the non-spherical shape of the spoon.
I have used the images from this website.
Best Answer
We see an inverted image in a spoon when kept far from our face with concave side towards our face. It is because our face is outside the focal legth of the conave side of the spoon.
We see a virtual inverted image whereas in case of concave mirror we can see a virtual image which is erect. This is because when the object is between pole and focus the we get erect enlarged image.
I tried to get a virtual image with a spoon, but I couldn't get even after bringing it too close to my face, this is probably because the focal length of the spoon was too small.
@HarnoorSingh Yes the virtual image of the nail is formed when close to the mirror it is because the nail is between the pole and the focus of the font side (concave) of the spoon, resulting in the formation of virtual image on the spoon. This situation is depicted by the second diagram.