  The Milky Way has accreted many ultra-faint dwarf galaxies (UFDs), and stars
from these galaxies can be found throughout our Galaxy today. Studying these
stars provides insight into galaxy formation and early chemical enrichment, but
identifying them is difficult. Clustering stellar dynamics in 4D phase space
($E$, $L_z$, $J_r$, $J_z$) is one method of identifying accreted structure
which is currently being utilized in the search for accreted UFDs. We produce
32 simulated stellar halos using particle tagging with the \textit{Caterpillar}
simulation suite and thoroughly test the abilities of different clustering
algorithms to recover tidally disrupted UFD remnants. We perform over 10,000
clustering runs, testing seven clustering algorithms, roughly twenty
hyperparameter choices per algorithm, and six different types of data sets each
with up to 32 simulated samples. Of the seven algorithms, HDBSCAN most
consistently balances UFD recovery rates and cluster realness rates. We find
that even in highly idealized cases, the vast majority of clusters found by
clustering algorithms do not correspond to real accreted UFD remnants and we
can generally only recover $6\%$ of UFDs remnants at best. These results focus
exclusively on groups of stars from UFDs, which have weak dynamic signatures
compared to the background of other stars. The recoverable UFD remnants are
those that accreted recently, $z_{\text{accretion}}\lesssim 0.5$. Based on
these results, we make recommendations to help guide the search for
dynamically-linked clusters of UFD stars in observational data. We find that
real clusters generally have higher median energy and $J_r$, providing a way to
help identify real vs. fake clusters. We also recommend incorporating chemical
tagging as a way to improve clustering results.
