  We study the correlations between parameters characterizing neutrino physics
and the evolution of dark energy. Using a fluid approach, we show that
time-varying dark energy models exhibit degeneracies with the cosmic neutrino
background over extended periods of the cosmic history, leading to a degraded
estimation of the total mass and number of species of neutrinos. We investigate
how to break degeneracies and combine multiple probes across cosmic time to
anchor the behaviour of the two components. We use Planck CMB data and BAO
measurements from the BOSS, SDSS and 6dF surveys to present current limits on
the model parameters, and then forecast the future reach from the CMB Stage-4
and DESI experiments. We show that a multi-probe analysis of current data
provides only marginal improvement on the determination of the individual
parameters and no reduction of the correlations. Future observations will
better distinguish the neutrino mass and preserve the current sensitivity to
the number of species even in case of a time-varying dark energy component.
