= Solution
For the <dielectric> half-space the two formal sources coincide at $(a,0,0)$, outside the region being represented. Their sum is $Q+Q'=2\epsilon_rQ/(1+\epsilon_r)$ in the <dielectric> Coulomb kernel, hence
$$
\boxed{\phi_-(x,y,z)=\frac{2Q}{4\pi\epsilon_0(1+\epsilon_r)r_+}\quad(x<0).}
$$
There is no charge singularity inside this half-space. The expression satisfies <Laplace's equation> there, decays at infinity, and matches the vacuum solution's tangential <electric field> and normal electric displacement. The coincident image is a mathematical representation, not an extra physical charge in the vacuum region.
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