  The ringdown signal emitted during a binary black hole coalescence can be
modeled as a linear superposition of the characteristic damped modes of the
remnant black hole that get excited during the merger phase. While checking the
consistency of the measured frequencies and damping times against the Kerr BH
spectrum predicted by General Relativity (GR) is a cornerstone of strong-field
tests of gravity, the consistency of measured excitation amplitudes and phases
have been largely left unexplored. For a nonprecessing, quasi-circular binary
black hole merger, we find that GR predicts a narrow region in the space of
mode amplitude ratio and phase difference, independently of the spin of the
binary components. % Using this unexpected result, we develop a new null test
of strong-field gravity which demands that the measured amplitudes and phases
of different ringdown modes should lie within this narrow region predicted by
GR. We call this the \emph{amplitude-phase consistency test} and introduce a
procedure for performing it using information from the ringdown signal. Lastly,
we apply this test to the GW190521 event, using the multimodal ringdown
parameters inferred by Capano et al. (2021) \cite{Capano:2021etf}. While
ringdown measurements errors for this event are large, we show that GW190521 is
consistent with the amplitude-phase consistency test. Our test is particularly
well suited for accommodating multiple loud ringdown detections as those
expected in the near future, and can be used complementarily to standard
black-hole spectroscopy as a proxy for modified gravity, compact objects other
than black holes, binary precession and eccentricity.
