Suppose that two structures $A$ and $B$ whose cardinality is greater than 1 (added in a revision) have the same positive primitive theory. Does it follow that the union of the full Horn theory of $A$ and that of $B$ is consistent? (Here I allow possibly empty finite conjunctions in the antecedent of a basic Horn formula and $\bot$ in the consequent; also a Horn sentence can have an arbitrarily many quantifiers.)
One way of having a contradiction from two Horn theories is having a sentence in one and its negation in the other. Such a sentence is positive primitive (or negations thereof). Hence the question in the beginning. In fact, it is true of any concrete structures that I can think of.
First of all, I stated a misunderstanding of mine in my original post: arbitrarily quantified literals also have as their negations Horn sentences. This explains Dr Alex Kruckman's examples on strict total orders and Professor Keith Kearnes's example on graphs.
Secondly, here is an example involving total orders in the language $\{\le\}$. Consider again naturals and integers. The chain of natural numbers satisfy the Horn sentence $(\exists x)(\forall y)[y \le x \to y = x]$, where as the chain of integers satisfy the Horn sentence $(\forall x)(\exists y)[y \not \ge x]$. Now the theory of partial orders is Horn, so it suffices to show that no partial order satisfies those two Horn sentences. This is easy, as the minimum element of a partial order, if it exists, cannot have an element strictly below it or an element incomparable with it.