Constitutive relations are formulated for a class of incompressible viscoelastic fluids for which internal structural changes occur at a rate that is influenced by the history of the strain. For the materials considered, the contribution to the stress at timetmade by the strain at an earlier time τ is a function of that strain, the true elapsed timet−τ,and a quantityσ(t,τ)that can be interpreted as the elapsed time measured by a clock whose rate of advance, because it is tied to the rate of structural change, is affected by the history of the strain. The functional relating∂σ(t,τ)/∂tto the history of the strain up to timetis assumed to have the same domain and a structure similar to that relating stress to strain history. The present theory reduces to the theory of BKZ fluids in the (extreme) special case in whichσ(t,τ)≡t−τ,in other words, in which∂σ(t,τ)/∂t≡1.It is shown that there is a sense in which constitutive relations recently found to account well for observed discrepancies between experimental observations and predictions of the BKZ theory can be considered first‐order approximations to the relations formulated here.