  An exploratory study of the time-like pion electromagnetic form factor in a
Poincar\'e-covariant bound state formalism in the isospin symmetric limit is
presented. Starting from a quark interaction kernel representing
gluon-intermediated interactions for valence-type quarks, non-valence effects
are included by introducing pions as explicit degrees of freedom. The two most
important qualitative aspects are, in view of the presented study, the opening
of the dominant $\rho$-meson decay channel and the presence of a multi-particle
branch cut setting in when the two-pion threshold is crossed. Based on a recent
respective computation of the quark-photon vertex, the pion electromagnetic
form factor for space-like and time-like kinematics is calculated. The obtained
results for its absolute value and its phase compare favorably to the available
experimental data, and they are analyzed in detail by confronting them to the
expectations based on an isospin-symmetric version of a vector-meson dominance
model.
