Given two complex analytic functions f and g on some compact domain $D$, I want to construct a new analytic function h on $D$ such that $$h(z) = 0 \iff f(z) = 0\ \text{ and }\ g(z) = 0$$ I want to do this without finding the zeros of $f$ and $g$. If I wanted to find $h$ such that $$ h(z) = 0 \iff f(z) = 0 \ \text{ or }\ g(z) = 0, $$ this can be easily done by letting $$ h(z) = f(z)g(z). $$
In a more specific version, we can assume that zeros of $f$ and $g$ lie on the real line. Thanks!
There is no formula of the form $$h(z)=F(f(z),g(z)).$$
Proof: Letting $f(z)=z$, $g(z)=c$ shows that $z\mapsto F(z,c)$ is holomorphic. Similarly for $F(c,w)$, so a theorem of Hartogs shows that $F$ is a holomorphic function of two variables. Now $F(0,0)=0$, but the zeroes of a holomorphic function in $\Bbb C^2$ cannot be isolated, by another theorem of Hartogs. So it cannot happen that $F(f(z),g(z))=0$ if and only if $f(z)=g(z)=0$.