  We quantify for the first time the gravitational wave (GW) phase shift
appearing in the waveform of eccentric binary black hole (BBH) mergers formed
dynamically in three-body systems. For this, we have developed a novel
numerical method where we construct a reference binary, by evolving the
post-Newtonian (PN) evolution equations backwards from a point near merger
without the inclusion of the third object, that can be compared to the real
binary that evolves under the influence from the third BH. From this we
quantify how the interplay between dynamical tides, PN-effects, and the
time-dependent Doppler shift of the eccentric GW source results in unique
observable GW phase shifts that can be mapped to the gravitational dynamics
taking place at formation. We further find a new analytical expression for the
GW phase shift, which surprisingly has a universal functional form that only
depends on the time-evolving BBH eccentricity. The normalization scales with
the BH masses and initial separation, which can be linked to the underlying
astrophysical environment. GW phase shifts from a chaotic 3-body BH scattering
taking place in a cluster, and from a BBH inspiraling in a disk migration trap
near a super-massive BH, are also shown for illustration. When current and
future GW detectors start to observe eccentric GW sources with high enough
signal-to-noise-ratio, we propose this to be among the only ways of directly
probing the dynamical origin of individual BBH mergers using GWs alone.
