  We identify a one-dimensional supersolid phase in a binary mixture of
near-hardcore bosons with weak, local inter-species repulsion. We find
realistic conditions under which such a phase, defined here as the coexistence
of quasi-superfluidity and quasi-charge density wave order, can be produced and
observed in finite ultra-cold atom systems in a harmonic trap. Our analysis is
based on Luttinger liquid theory supported with numerical calculations using
the time-evolving block decimation method. Clear experimental signatures of
these two orders can be found, respectively, in time-of-flight interference
patterns, and the structure factor S(k) derived from density correlations.
