  We study ghost distributions on supersymmetric spaces for the case of basic
classical Lie superalgebras. We introduce the notion of interlaced pairs, which
are those for which both $(\mathfrak{g},\mathfrak{k})$ and
$(\mathfrak{g},\mathfrak{k}')$ admit Iwasawa decompositions. For such pairs we
define a ghost algebra, generalizing the subalgebra of
$\mathcal{U}\mathfrak{g}$ defined by Gorelik. We realize this algebra as an
algebra of $G$-equivariant operators on the supersymmetric space itself, and
for certain pairs, the `special' ones, we realize our operators as
twisted-equivariant differential operators on $G/K$. We additionally show that
the Harish-Chandra morphism is injective, compute its image for all rank one
pairs, and provide a conjecture for the image when
$(\mathfrak{g},\mathfrak{k})$ is interlaced.
