  Ultra-high energy neutrinos are detectable through impulsive radio signals
generated through interactions in dense media, such as ice. Subsurface in-ice
radio arrays are a promising way to advance the observation and measurement of
astrophysical high-energy neutrinos with energies above those discovered by the
IceCube detector ($\geq$1 PeV) as well as cosmogenic neutrinos created in the
GZK process ($\geq$100 PeV). Here we describe the $\textit{NuPhase}$ detector,
which is a compact receiving array of low-gain antennas deployed 185 m deep in
glacial ice near the South Pole. Signals from the antennas are digitized and
coherently summed into multiple beams to form a low-threshold interferometric
phased array trigger for radio impulses. The NuPhase detector was installed at
an Askaryan Radio Array (ARA) station during the 2017/18 Austral summer season.
$\textit{In situ}$ measurements with an impulsive, point-source calibration
instrument show a 50% trigger efficiency on impulses with voltage
signal-to-noise ratios (SNR) of $\le$2.0, a factor of $\sim$1.8 improvement in
SNR over the standard ARA combinatoric trigger. Hardware-level simulations,
validated with $\textit{in situ}$ measurements, predict a trigger threshold of
an SNR as low as 1.6 for neutrino interactions that are in the far field of the
array. With the already-achieved NuPhase trigger performance included in
ARASim, a detector simulation for the ARA experiment, we find the trigger-level
effective detector volume is increased by a factor of 1.8 at neutrino energies
between 10 and 100 PeV compared to the currently used ARA combinatoric trigger.
We also discuss an achievable near term path toward lowering the trigger
threshold further to an SNR of 1.0, which would increase the effective
single-station volume by more than a factor of 3 in the same range of neutrino
energies.
