  We provide a detailed analysis of how bosonic dark matter "condensates"
interact with compact stars, extending significantly the results of a recent
Letter. We focus on bosonic fields with mass $m_B$, such as axions, axion-like
candidates and hidden photons. Self-gravitating bosonic fields generically form
"breathing" configurations, where both the spacetime geometry and the field
oscillate, and can interact and cluster at the center of stars. We construct
stellar configurations formed by a perfect fluid and a bosonic condensate, and
which may describe the late stages of dark-matter accretion onto stars, in dark
matter-rich environments. These composite stars oscillate at a frequency which
is a multiple of $f=2.5\times 10^{14}\,\left(m_{B}c^2/eV\right)\,{\rm Hz}$.
Using perturbative analysis and Numerical Relativity techniques, we show that
these stars are generically stable, and we provide criteria for instability.
Our results also indicate that the growth of the dark matter core is halted
close to the Chandrasekhar limit. We thus dispel a myth concerning dark matter
accretion by stars: dark mater accretion does not necessarily lead to the
destruction of the star, nor to collapse to a black hole. Finally, we argue
that stars with long-lived bosonic cores may also develop in other theories
with effective mass couplings, such as (massless) scalar-tensor theories.
