  One of the most exciting targets of current and future gravitational-wave
observations is the angular power spectrum of the astrophysical GW background.
This cumulative signal encodes information about the large-scale structure of
the Universe, as well as the formation and evolution of compact binaries
throughout cosmic time. However, the finite rate of compact binary mergers
gives rise to temporal shot noise, which introduces a significant bias in
measurements of the angular power spectrum if not explicitly accounted for.
Previous work showed that this bias can be removed by cross-correlating GW sky
maps constructed from different observing times. However, this work considered
an idealised measurement scenario, ignoring detector specifics and in
particular noise contributions. Here we extend this temporal cross-correlation
method to account for these difficulties, allowing us to implement the first
unbiased anisotropic search pipeline for LIGO-Virgo-KAGRA data. In doing so, we
show that the existing pipeline is biased even in the absence of shot noise,
due to previously neglected sub-leading contributions to the noise covariance.
We apply our pipeline to mock LIGO data, and find that our improved analysis
will be crucial for stochastic searches from the current observing run (O4)
onwards.
