  $\omega$-clones are multi-sorted structures that naturally emerge as algebras
for infinite trees, just as $\omega$-semigroups are convenient algebras for
infinite words. In the algebraic theory of languages, one hopes that a language
is regular if and only if it is recognized by an algebra that is finite in some
simple sense. We show that, for infinite trees, the situation is not so simple:
there exists an $\omega$-clone that is finite on every sort and finitely
generated, but recognizes a non-regular language.
