  We compute the correlated electronic structure of stacked $1T$-TaS$_2$
bilayers using the $GW$ + EDMFT method. Depending on the surface termination,
the semi-infinite uncorrelated system is either band-insulating or exhibits a
metallic surface state. For realistic values of the onsite and intersite
interactions, a Mott gap opens in the surface state, but this gap is smaller
than the gap originating from the bilayer structure. Our results are consistent
with recent scanning tunneling spectroscopy measurements for different
terminating layers, and with our own photoemission measurements, which indicate
the coexistence of spatial regions with different gaps in the electronic
spectrum.
