  We study three different extended scalar-tensor theories of gravity by also
allowing a negative sign for the kinetic term for the scalar field in the
Jordan frame. Our scope is to understand how the observational constraints for
these models cope with the volume of the parameter space in which the theory is
healthy. Models with a negative kinetic term lead to decreasing effective
gravitational constant with redshift and behave as an effective relativistic
component with a negative energy density as opposite to their corresponding
version with a standard kinetic term. As a consequence, we find that the
extended branch with a negative sign for the kinetic term correspond in general
to lower $H_0$ and $\sigma_8$ compared to $\Lambda$CDM. We find that in all the
cases with a negative sign for the kinetic term studied here, cosmological
observations constrain these models around GR and prefer a volume of the
parameter space in which the theory is not healthy since the scalar field
behave as a ghost also in the related Einstein frame. We show that also in the
phantom branch early modify gravity with a quartic coupling can substantially
reduce the $H_0$ tension fitting the combination of cosmic microwave background
data from Planck, baryon acoustic oscillations from BOSS and eBOSS, and
Supernovae from the Pantheon sample with calibration information by SH0ES.
