I know that the question may look similar to this: Is fibre product of varieties irreducible (integral)?, but I am forced by the context to use a different definition for variety.
Definition. Let $K$ be a field (not necessarly algebraically closed), and $X$ a scheme over $K$. Then $X$ is a variety if it is separated, of finite type, and geometrically integral.
Assume that $X$ and $Y$ are two varieties. Is it true that $X\times_K Y$ (the scheme-theoretical fiber product) is a variety? (Of course in the sense of the previous definition).
If you are able to find a clear reference for this, it would be better than an explicit argument.