  Decay of bound states due to coupling with free particle states is a general
phenomenon occurring at energy scales from MeV in nuclear physics to peV in
ultracold atomic gases. Such a coupling gives rise to Fano-Feshbach resonances
(FFR) that have become key to understanding and controlling interactions - in
ultracold atomic gases, but also between quasiparticles such as microcavity
polaritons. The energy positions of FFR were shown to follow quantum chaotic
statistics. In contrast, lifetimes which are the fundamental property of a
decaying state, have so far escaped a similarly comprehensive understanding.
Here we show that a bound state, despite being resonantly coupled to a
scattering state, becomes protected from decay whenever the relative phase is a
multiple of $\pi$. We observe this phenomenon by measuring lifetimes spanning
four orders of magnitude for FFR of spin-orbit excited molecular ions with
merged beam and electrostatic trap experiments. Our results provide a blueprint
for identifying naturally long-lived states in a decaying quantum system.
