  Using parallaxes from Gaia Early Data Release 3 (EDR3), we determine
multi-wavelength BVI, JHK, and [3.6] & [4.5] micron absolute magnitudes for 37
nearby Milky Way Cepheids, covering the period range between 5 and 60 days. We
apply these period-luminosity relations to Cepheids in the Large and Small
Magellanic Clouds, and find that the derived distances are significantly
discrepant with the geometric distances according to detached eclipsing
binaries (DEBs). We explore several potential causes of these issues, including
reddening, metallicity, and the existence of an additional zero-point offset,
but none provide a sufficient reconciliation with both DEB distances. We
conclude that the combination of the systematic uncertainties on the EDR3
parallaxes with the uncertainties on the effect of metallicity on the Cepheid
distance scale leads to a systematic error floor of approximately 3%. We
therefore find that the EDR3 data is not sufficiently accurate in the regime of
these bright Cepheids to determine extragalactic distances precise to the 1%
level at this time, in agreement with a number of contemporary studies.
