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We extend ${\mathcal H}_1$-clauses with disequalities between images of terms under a tree homomorphism (hom-disequalities). This extension allows to test whether two terms are distinct modulo a semantic interpretation, allowing, e.g., to neglect information that is not considered relevant for the intended comparison. We prove that ${\mathcal H}_1$-clauses with hom-disequalities are more expressive than ${\mathcal H}_1$-clauses with ordinary term disequalities, and that they are incomparable with ${\mathcal H}_1$-clauses with disequalities between paths. Our main result is that ${\mathcal H}_1$-clauses with this new type of constraints can be normalized into an equivalent tree automaton with hom-disequalities. Since emptiness for that class of automata turns out to be decidable, we conclude that satisfiability is decidable for positive Boolean combinations of queries to predicates defined by ${\mathcal H}_1$-clauses with hom-disequalities.