  We study a certain compactification of the Drinfeld period domain over a
finite field which arises naturally in the context of Drinfeld moduli spaces.
Its boundary is a disjoint union of period domains of smaller rank, but these
are glued together in a way that is dual to how they are glued in the
compactification by projective space. This compactification is normal and
singular along all boundary strata of codimension $\ge2$. We study its geometry
from various angles including the projective coordinate ring with its Hilbert
function, the cohomology of twisting sheaves, the dualizing sheaf, and give a
modular interpretation for it. We construct a natural desingularization which
is smooth projective and whose boundary is a divisor with normal crossings. We
also study its quotients by certain finite groups.
