A tree automaton with global equality and disequality constraints, TAGED for short, is an automaton on trees which allows to test (dis)equalities between subtrees which may be arbitrarily faraway. In particular, it is equipped with an (dis)equality relation on states, so that whenever two subtrees t and t' evaluate (in an accepting run) to two states which are in the (dis)equality relation, they must be (dis)equal. We study several properties of TAGEDs, and prove decidability of emptiness of several classes. We give two applications of TAGEDs: decidability of an extension of Monadic Second Order Logic with tree isomorphism tests and of unification with membership constraints.