  We develop an approximation-free Diagrammatic Monte Carlo technique to study
fermionic particles interacting with each other simultaneously through both an
attractive Coulomb potential and bosonic excitations of the underlying medium.
Exemplarily we apply the method to the long-standing exciton-polaron problem
and present numerically exact results for the wave function, ground-state
energy, binding energy and effective mass of this quasiparticle. Focusing on
the electron-hole pair bound-state formation, we discuss various limiting cases
of a generic exciton-polaron model. The frequently used instantaneous
approximation to the retarded interaction due to the phonon exchange is found
to be of very limited applicability. For the case of a light electron and heavy
hole the system is well approximated by a particle in the field of a static
attractive impurity.
