What is “Real coordinate space”?

What is the Real Coordinate Space in the discussion of vectors?

How does it relate to Cartesian Coordinate System and Euclidean Space?

In my experience the expression "real coordinate space" emphasizes that we are not working over the complex numbers, i.e. the space is $\mathbb R^n$, not $\mathbb C^n$. You can use Cartesian coordinates (and a whole bunch of other coordinate systems) on these spaces. The spaces $\mathbb R^n$ are called Euclidean spaces, so they are the same as real coordinate spaces.