Suppose that $V$, $W_1$, $W_2$ are smooth, irreducible varieties and $f_i:W_i\rightarrow V$ is a surjective rational map with finite fibres. In general, the corresponding fibre product $W_1\times_V W_2$ is not irreducible. For example, if $W_1$ is the curve $y^2-x=0$ and $W_1$ is the curve $z^2+x=0$, then the fibre product $W_1\times_{\mathbb{A}^1}W_2$ is the union of the varieties $\{y^2-x, z-iy\}$ and $\{y^2-x, z+iy\}$.
I was wondering if there is some general criterion for when such a fibre product is irreducible.
What if the maps $f_i$ are just projections?