  We present an analogue spacetime model that reproduces the salient features
of the most common ansatz for quantum gravity phenomenology. We do this by
investigating a system of two coupled Bose-Einstein condensates. This system
can be tuned to have two "phonon" modes (one massive, one massless) which share
the same limiting speed in the hydrodynamic approximation [Phys. Rev. D72
(2005) 044020, gr-qc/0506029; cond-mat/0409639]. The system nevertheless
possesses (possibly non-universal) Lorentz violating terms at very high
energies where "quantum pressure" becomes important. We investigate the
physical interpretation of the relevant fine-tuning conditions, and discuss the
possible lessons and hints that this analogue spacetime could provide for the
phenomenology of real physical quantum gravity. In particular we show that the
effective field theory of quasi-particles in such an emergent spacetime does
not exhibit the so called "naturalness problem".
