Multivariate Complex Function

Question

Suppose $f(x,w)\not=0$ for all $x,w\in H^+\cup H^-$ (open upper and lower half planes) and $f$ is a multivariate entire function. Must there exist univariate entire functions $\phi_1$ and $\phi_2$ with only real zeros, and multivariate entire function $g$ such that $$f(x,w)=\phi_1(x)\phi_2(w)e^{g(x,w)}?$$

Answer 1

1) Not sure what you are aiming for here.

2) Sure. Take for example $f(x,w) = xwe^{x+w}$

3) No. See 2)