  Following the recommendations to NASA and ESA, the search for life on
exoplanets will be a priority in the next decades. Two direct imaging space
mission concepts are being developed: the Habitable Worlds Observatory (HWO)
and the Large Interferometer for Exoplanets (LIFE). HWO focuses on reflected
light spectra in the ultraviolet/visible/near-infrared (UV/VIS/NIR), while LIFE
captures the mid-infrared (MIR) emission of temperate exoplanets. We assess the
potential of HWO and LIFE in characterizing a cloud-free Earth twin orbiting a
Sun-like star at 10 pc, both separately and synergistically, aiming to quantify
the increase in information from joint atmospheric retrievals on a habitable
planet. We perform Bayesian retrievals on simulated data from an HWO-like and a
LIFE-like mission separately, then jointly, considering the baseline spectral
resolutions currently assumed for these concepts and using two increasingly
complex noise simulations. HWO would constrain H$_2$O, O$_2$, and O$_3$, in the
atmosphere, with ~ 100 K uncertainty on the temperature profile. LIFE would
constrain CO$_2$, H$_2$O, O$_3$ and provide constraints on the thermal
atmospheric structure and surface temperature (~ 10 K uncertainty). Both
missions would provide an upper limit on CH$_4$. Joint retrievals on HWO and
LIFE data would accurately define the atmospheric thermal profile and planetary
parameters, decisively constrain CO$_2$, H$_2$O, O$_2$, and O$_3$, and weakly
constrain CO and CH$_4$. The detection significance is greater or equal to
single-instrument retrievals. Both missions provide specific information to
characterize a terrestrial habitable exoplanet, but the scientific yield is
maximized with synergistic UV/VIS/NIR+MIR observations. Using HWO and LIFE
together will provide stronger constraints on biosignatures and life
indicators, potentially transforming the search for life in the universe.
