  We consider the screening of the axio-dilaton fields when both the dilaton
and the axion couple to matter with Yukawa couplings. We analyse the screening
of the dilaton in the vicinity of a compact object and find that this can only
take place when special boundary conditions at infinity are imposed. We study
the cosmological dynamics of the axio-dilaton system when coupled to matter
linearly and find that the special boundary conditions at infinity, which
guarantee the screening of compact objects, do not generically emerge from
cosmology. We analyse the background cosmology and the cosmological
perturbations at late time in these models and show that the Baryon Acoustic
Oscillations constrain the coupling of the dilaton to matter to be smaller than
in its natural supergravity realisation. Moreover we find that the Hubble rate
in the present Universe could deviate from the normalised Planck value,
although by an amount too small to account for the $H_0$ tension, and that the
growth of structure is generically reduced compared to $\Lambda$CDM.
