  Fermion soliton stars are a motivated model of exotic compact objects in
which a nonlinear self-interacting real scalar field couples to a fermion via a
Yukawa term, giving rise to an effective fermion mass that depends on the fluid
properties. Here we continue our investigation of this model within General
Relativity by considering a scalar potential with generic asymmetric vacua.
This case provides fermion soliton stars with a parametrically different
scaling of the maximum mass relative to the model parameters, showing that the
special case of symmetric vacua, in which we recover our previous results,
requires fine tuning. In the more generic case studied here the mass and radius
of a fermion soliton star are comparable to those of a neutron star for natural
model parameters at the GeV scale. Finally, the asymmetric scalar potential
inside the star can provide either a positive or a negative effective
cosmological constant in the interior, being thus reminiscent of gravastars or
anti-de Sitter bubbles, respectively. In the latter case we find the existence
of multiple, disconnected, branches of solutions.
