  We study the low-temperature phases of interacting bosons on a
two-dimensional quasicrystalline lattice. By means of numerically exact Path
Integral Monte Carlo simulations, we show that for sufficiently weak
interactions the system is a homogeneous Bose-Einstein condensate, which
develops density modulations for increasing filling factor. The simultaneous
occurrence of sizeable condensate fraction and density modulation can be
interpreted as the analogous, in a quasicrystalline lattice, of supersolid
phases occurring in conventional periodic lattices. For sufficiently large
interaction strength and particle density, global condensation is lost and
quantum exchanges are restricted to specific spatial regions. The emerging
quantum phase is therefore a Bose Glass, which here is stabilized in the
absence of any source of disorder or quasidisorder, purely as a result of the
interplay between quantum effects, particle interactions and quasicrystalline
substrate. This finding clearly indicates that (quasi)disorder is not essential
to observe Bose Glass physics. Our results are of interest for ongoing
experiments on (quasi)disorder-free quasicrystalline lattices.
