  We construct several new examples of homogeneous domains in complex space
that do not have bounded realisations. They are equivalent to tubes over
affinely homogeneous domains in real space and have a real-analytic everywhere
Levi non-degenerate non-umbilic boundary. Using the geometry of the boundary,
we determine the full automorphism groups of the domains. We also discuss some
interesting examples of tube domains with everywhere umbilic boundary (i.e.,
boundary equivalent to the corresponding quadric).
