  High-precision measurements are crucial for testing the fundamental laws of
nature and for advancing the technological frontier. Clock interferometry,
where particles with an internal clock are coherently split and recombined
along two spatial paths, has sparked significant interest due to its
fundamental implications, especially at the intersection of quantum mechanics
and general relativity. Here, we demonstrate that a clock interferometer
provides metrological improvement with respect to its technical-noise-limited
counterpart employing a single internal quantum state. This enhancement around
a critical working point can be interpreted as a geometric-phase-induced
signal-to-noise ratio gain. In our experimental setup, we infer a precision
enhancement of 8.8 decibels when measuring a small difference between external
fields. We estimate that tens of decibels of precision enhancement could be
attained for measurements with a higher atom flux. This opens the door to the
development of a superior probe for fundamental physics as well as a
high-performance sensor for various technological applications.
