The empty signature still needs a nonempty term domain

With no constants or functions, the term constructors give only variables, so none is closed. In the expanded signature each closed term is exactly one of the constants $c_n$, because there are still no function symbols. Let $A$ have carrier $\{0\}$ and interpret every $c_n$ by $0$. Put $H=\mathrm{Th}(A)$, the set of all expanded sentences true in this structure. Negation's truth clause makes exactly one of $\sigma ,\mathrm{\neg }\sigma$ belong to $H$. F2 makes $H$ consistent; it also makes $H$ deductively closed, since any sentence proved from $H$ is true in $A$.

For any existential sentence $\mathrm{\exists }x\varphi$, if it is true in $A$, its only possible witness is $0=c_0^A$, so $\varphi [c_0\mathrm{/}x]$ is true. If it is false, the implication $\mathrm{\exists }x\varphi \to \varphi [c_0\mathrm{/}x]$ is true by the Boolean implication clause. Thus every such witness axiom belongs to $H$, and $H$ is Henkin with seed $c_0$. For all $m,n$ the equation $c_m=c_n$ is true since $0=0$, so all and only the closed terms are in the one class $[c_0]=\{c_n:n<\omega \}$. The quotient of F1 is exactly $\{[c_0]\}$. Its reduct is a singleton model of the original empty theory. This computation concerns this particular $H$, not every Henkin completion of the empty theory.