Does the following non-separable wavefunction represent an entangled state?

$\psi(x1,x2)$ = $\exp[i b x_{1}x_{2}]\phi(x_{1})\phi(x_{2})$

This state can not be factorized into functions of $x_{1}$ and $x_{2}$, but one might argue that an overall phase factor is physically irrelevant and only consider $\phi(x_{1})\phi(x_{2})$, is it valid to ignore the phase factor in this case?

Does the following non-separable wavefunction represent an entangled state?

$\psi(x_1,x_2)$ = $\exp[i b x_{1}x_{2}]\phi(x_{1})\phi(x_{2})$

This state can not be factorized into functions of $x_{1}$ and $x_{2}$, but one might argue that an overall phase factor is physically irrelevant and only consider $\phi(x_{1})\phi(x_{2})$, is it valid to ignore the phase factor in this case?