Let's consider the following theory:
$$L= -\frac{1}{4}F_{\mu \nu}F^{\mu\nu} +{1\over 2} |D_\mu \Phi|^2 +{1\over 2}|D_\mu \chi|^2 + \lambda_1\bigl(|\Phi|^2-\frac{v_1^2}{2}\bigr) +\lambda_2\bigl(|\chi|^2-\frac{v_2^2}{2}\bigr)$$
where $\Phi$ and $\chi$ are complex scalars coupled to a $U(1)$ gauge boson $A_\mu$ through the usual covariant derivative: $$D_\mu \Phi= (\partial_\mu -ieA_\mu)\Phi $$ $$D_\mu \chi= (\partial_\mu -ieA_\mu)\chi $$
Expanding each scalar around its vev, we find $$L= \dots + \frac{1}{2}e^2(v_1+h_1(x))^2(A_\mu-\frac{1}{ev_1}\partial_\mu\xi_1(x))^2 +\frac{1}{2}e^2(v_2+h_2(x))^2(A_\mu-\frac{1}{ev_2}\partial_\mu\xi_2(x))^2$$
Where $h_i(x)$ are the Higgs-like bosons and $\xi_i(x)$ the respective Goldstone bosons. How must gauge invariance be used in this case to reproduce the unitary gauge?