  Current instruments and spectral analysis programs are now able to decompose
the integrated spectrum of a stellar system into distributions of ages and
metallicities. The reliability of these methods have rarely been tested on
nearby systems with resolved stellar ages and metallicities. Here we derive the
age-metallicity distribution of M54, the nucleus of the Sagittarius dwarf
spheroidal galaxy, from its integrated MUSE spectrum. We find a dominant old
(8-14 Gyr), metal-poor (-1.5 dex) and a young (1 Gyr), metal-rich (+0.25 dex)
component - consistent with the complex stellar populations measured from
individual stars in the same MUSE data set. There is excellent agreement
between the (mass-weighted) average age and metallicity of the resolved and
integrated analyses. Differences are only 3% in age and 0.2 dex metallicitiy.
By co-adding individual stars to create M54's integrated spectrum, we show that
the recovered age-metallicity distribution is insensitive to the magnitude
limit of the stars or the contribution of blue horizontal branch stars - even
when including additional blue wavelength coverage from the WAGGS survey.
However, we find that the brightest stars can induce the spurious recovery of
an old ($>8$ Gyr), metal-rich (+0.25 dex) stellar population, which is
otherwise not expected from our understanding of chemical enrichment in M54.
The overall derived stellar mass-to-light ratio of M54 is
M/L$_{\mathrm{V}}=1.46$ with a scatter of 0.22 across the field-of-view, which
we attribute to the stochastic contribution of a young, metal-rich component.
These findings provide strong evidence that complex stellar population
distributions can be reliably recovered from integrated spectra of
extragalactic systems.
