= Solution
Use a <first-order signature> with a constant $a$ and unary predicates $P,Q$, and set
$$
F=\forall x\,[P(x)\lor Q(x)],
\qquad G=\exists x\,[P(x)\lor Q(x)].
$$
The displayed <first-order clause> has two positive atoms, so it is not a <Horn clause>. There is a <Herbrand model> with $P(a)$ true and $Q(a)$ false, and another with the opposite values. Thus neither atom holds in all <Herbrand models>; the intersection construction makes both false.
Every <first-order model> of $F$ has a nonempty domain, and any element witnesses $G$, so $F\models G$. Nevertheless $\mathcal H_F\not\models G$. Therefore
$$
\boxed{F\models G\quad\text{but}\quad\mathcal H_F\not\models G.}
$$
This example uses a positive <existential sentence>, so it also shows the necessity of the <Horn clause> restriction in the repaired result from part (d). Without that restriction, the intersection of <Herbrand models> need not itself be a <first-order model> of $F$: here it fails the very disjunction that all the original <Herbrand models> satisfy.
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