  We study the (pseudo-) anyon Hubbard model on a one-dimensional lattice
without the presence of a three-body hardcore constraint. In particular, for
the pseudo-fermion limit of a large statistical angle $\theta\approx\pi$, we
observe a wealth of exotic properties including {a first order transition}
between different superfluid phases and a {two-component} partially paired
phase for large fillings without need of an additional three-body hardcore
constraint.In this limit, we analyze the effect of an induced hardcore
constraint, which leads to the stabilization of superfluid {ground states} for
vanishing or even small attractive on-site interactions. For finite statistical
angles, we study the unconventional broken-symmetry superfluid peaked at a
finite momentum, resulting in an interesting beat phenomenon of single particle
correlation functions.We show how some features of various ground state phases,
including an analog of the partially paired phase in the pseudo-fermion limit,
may be reproduced in a naive mean field frame.
