Let $X$, $Y$, $Z$ be integral schemes of finite type over a field $K$ (i.e., locally affine opens are finitely-generated algebras over $K$). Suppose we have the following condition$\colon$
 
$f \colon Y \to X$ is finite.

We consider a dominant morphism $g \colon Z \to X$. 

Q. Suppose that ${\cal O}_Y \cap {\cal O}_Z = {\cal O}_X$. Then is it possble that $Y \times_X Z$ is *not* integral while $Y \times_X \eta_Z$ being integral ?
-

As in the answer below in the case $Y = Z$, there is an example as $Y$ being a normalisation of $X$.