  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 the \'etale cohomology groups (with
Q_ell 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.
