  Characterizing the epoch of reionization (EoR) at $z\gtrsim 6$ via the
redshifted 21 cm line of neutral Hydrogen (HI) is critical to modern
astrophysics and cosmology, and thus a key science goal of many current and
planned low-frequency radio telescopes. The primary challenge to detecting this
signal is the overwhelmingly bright foreground emission at these frequencies,
placing stringent requirements on the knowledge of the instruments and
inaccuracies in analyses. Results from these experiments have largely been
limited not by thermal sensitivity but by systematics, particularly caused by
the inability to calibrate the instrument to high accuracy. The interferometric
bispectrum phase is immune to antenna-based calibration and errors therein, and
presents an independent alternative to detect the EoR HI fluctuations while
largely avoiding calibration systematics. Here, we provide a demonstration of
this technique on a subset of data from the Hydrogen Epoch of Reionization
Array (HERA) to place approximate constraints on the brightness temperature of
the intergalactic medium (IGM). From this limited data, at $z=7.7$ we infer
"$1\sigma$" upper limits on the IGM brightness temperature to be $\le 316$
"pseudo" mK at $\kappa_\parallel=0.33$ "pseudo" $h$ Mpc$^{-1}$ (data-limited)
and $\le 1000$ "pseudo" mK at $\kappa_\parallel=0.875$ "pseudo" $h$ Mpc$^{-1}$
(noise-limited). The "pseudo" units denote only an approximate and not an exact
correspondence to the actual distance scales and brightness temperatures. By
propagating models in parallel to the data analysis, we confirm that the
dynamic range required to separate the cosmic HI signal from the foregrounds is
similar to that in standard approaches, and the power spectrum of the
bispectrum phase is still data-limited (at $\gtrsim 10^6$ dynamic range)
indicating scope for further improvement in sensitivity as the array build-out
continues.
