  We study the shape and texture of finite domains comprising chiral/achiral
molecules carrying a tilt field embedded in a 2-dimensional surface. Using a
combination of simulations and {\it exact} variational calculations, we
determine the equilibrium (mean field) phase diagram in the achiral and chiral
cases. We find a variety of novel shapes and textures including a {\it
spontaneously broken chiral texture} when the molecules are achiral. We show
that chiral tilt-domains nucleating in a region of two-phase coexistence, {\it
repel} each other, thereby preventing coalescence and further growth. Our work
has implications for tilt texture domains in phospholipid monolayers and giant
unilamellar vesicles (GUVs), nucleating domains of Sm-C$^*$ in Sm-A films, and
chiral emulsions in Sm-A films.
