An irreducible scheme whose extension of scalars is connected and not irreducible.

Question

I have difficulty finding an example of an affine scheme such that satisfies

1 It is an affine scheme over the real number.

2 It is irreducible.

3 Its extension of scalers $\Bbb R\to \Bbb C$ is connected and not irreducible.

I have spent some time on it. Thanks for any answers, hint or references.

Answer 1

How about $\mathbb{R}[x,y]/(x^2 + y^2)$ ?