Scrutinizing the scattering of ordinarymesons in the limiting case of the number of colour degrees of freedomNc of quantumchromodynamics approaching infinity, we formulate Feynman-diagram selection criteria and from these deduce rigorous self-consistency conditions for the manifestation of a tetraquark, a two-quark–two-antiquark bound state, as a pole in the corresponding amplitudes. Our constraints bear rather far-reaching consequences: In particular, all flavour-exotic tetraquarks, composed of four (anti)quarks of disparate flavour, must come in, at least, two variants differing in (and thus readily identifiable by) the large-Nc behaviour of their couplings to two ordinary mesons. Quite generally, irrespective of their flavour composition, all tetraquarks prove to be narrow. Their decay rates behave, for large $N_c$, like $1/N_c^2$ and thus decrease faster than those of ordinary mesons.