  ${\bf Background:}$ Weakly bound and unbound nuclear states appearing around
particle thresholds are prototypical open quantum systems. Theories of such
states must take into account configuration mixing effects in the presence of
strong coupling to the particle continuum space.
  ${\bf Purpose:}$ To describe structure and decays of three-body systems, we
developed a Gamow coupled-channel (GCC) approach in Jacobi coordinates by
employing the complex-momentum formalism. We benchmarked the new framework
against the complex-energy Gamow Shell Model (GSM).
  ${\bf Methods:}$ The GCC formalism is expressed in Jacobi coordinates, so
that the center-of-mass motion is automatically eliminated. To solve the
coupled-channel equations, we use hyperspherical harmonics to describe the
angular wave functions while the radial wave functions are expanded in the
Berggren ensemble, which includes bound, scattering and Gamow states.
  ${\bf Results:}$ We show that the GCC method is both accurate and robust. Its
results for energies, decay widths, and nucleon-nucleon angular correlations
are in good agreement with the GSM results.
  ${\bf Conclusions:}$ We have demonstrated that a three-body GSM formalism
explicitly constructed in cluster-orbital shell model coordinates provides
similar results to a GCC framework expressed in Jacobi coordinates, provided
that a large configuration space is employed. Our calculations for $A=6$
systems and $^{26}$O show that nucleon-nucleon angular correlations are
sensitive to the valence-neutron interaction. The new GCC technique has many
attractive features when applied to bound and unbound states of three-body
systems: it is precise, efficient, and can be extended by introducing a
microscopic model of the core.

Nuclear Theory