= Solution
Put $\alpha_1=\alpha$, $\alpha_2=1-\alpha$, and introduce a categorical class label for each observation, encoded by $Z_{i1},Z_{i2}\in\{0,1\}$ with $Z_{i1}+Z_{i2}=1$. Its prior class <probability> is $\alpha_j$, and conditional on class $j$ the observation <probability density function> is $f_j(x_i)$. Thus its joint observation-label <probability density function> is
$$
p_\theta(x_i,Z_i)=\prod_{j=1}^2[\alpha_j f_j(x_i)]^{Z_{ij}}.
$$
Independence across observations yields the <finite Gaussian mixture with a common variance> complete-data <likelihood function>
$$
\boxed{L_c(\theta;x,Z)=\prod_{i=1}^n\prod_{j=1}^2[\alpha_j f_j(x_i)]^{Z_{ij}}.}
$$
Summing over the two possible labels at each observation recovers $\prod_i[\alpha f_1(x_i)+(1-\alpha)f_2(x_i)]$, the observed-data <likelihood function>. The product requested in the PDF is therefore a joint complete-data <likelihood function>. Strictly conditioning on $Z$ would remove the factors $\alpha_j$; the PDF's conditional notation for this joint product is imprecise.
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