  In this paper we introduce two new generalizations of Krull domains:
$\ast$-almost independent rings of Krull type ($\ast$-almost IRKTs) and
$\ast$-almost generalized Krull domains ($\ast$-AGKDs), neither of which need
be integrally closed. We characterize them using certain types of
$\ast$-homogeneous ideals. To do this we introduce $\ast$-almost
super-homogeneous ideals and $\ast$-almost super-SH domains. We prove that a
domain $D$ is a $\ast$-almost IRKT if and only if $D$ is a $\ast$-almost
super-SH domain and that a domain is a $\ast$-AGKD if and only if $D$ is a type
1 $\ast$-almost super-SH domain. Further, we study $\ast$-almost factorial
general-SH domains ($\ast$-afg SH domains) and we prove that a domain $D$ is a
$\ast$-afg-SH domain if and only if $D$ is a $\ast$-IRKT and an AGCD-domain.
