The chiral anomaly is a striking signature of quantum effects which lead to the non-conservation of a classically conserved current, specifically the chiral currents in systems of fermions. In condensed matter systems, the chiral anomaly can be realized in Weyl semimetals, which then exhibit a signature electromagnetic response associated to anomaly due to the separation of the Weyl points in momentum space. In the presence of strong interactions however, a Weyl semimetal phase can give rise to an ordered phase, and spontaneously break the chiral symmetry. This then leads to a Goldstone mode which can have intrinsic dynamics and fluctuations, leading to a dynamical chiral anomaly response -- a situation known as a dynamical axion insulator. Here we consider a simple model of this dynamical axion insulator and calculate the equations of motion for the Goldstone mode. Surprisingly, we find that the Goldstone mode appears to exhibit a negative phase stiffness, signalling a further instability of the system towards finite momentum. This is expected to lead to very strong fluctuations of the anomalous response. We suggest a long-wavelength theory of Lifschitz type which may govern the axion dynamics in this system and comment on possible signatures of this model.