We consider two K3 surfaces defined over an arbitrary field, together with a smooth proper moduli space of stable sheaves on each. When the moduli spaces have the same dimension, we prove that if thé etale cohomology groups (with Q ℓ coefficients) of the two surfaces are isomorphic as Galois representations, then the same is true of the two moduli spaces. In particular, if the field of definition is finite and the K3 surfaces have equal zeta functions, then so do the moduli spaces, even when the moduli spaces are not birational.