  A joint determination of the reactor antineutrino spectra resulting from the
fission of $^{235}$U and $^{239}$Pu has been carried out by the Daya Bay and
PROSPECT collaborations. This Letter reports the level of consistency of
$^{235}$U spectrum measurements from the two experiments and presents new
results from a joint analysis of both data sets. The measurements are found to
be consistent. The combined analysis reduces the degeneracy between the
dominant $^{235}$U and $^{239}$Pu isotopes and improves the uncertainty of the
$^{235}$U spectral shape to about 3\%. The ${}^{235}$U and $^{239}$Pu
antineutrino energy spectra are unfolded from the jointly deconvolved reactor
spectra using the Wiener-SVD unfolding method, providing a data-based reference
for other reactor antineutrino experiments and other applications. This is the
first measurement of the $^{235}$U and $^{239}$Pu spectra based on the
combination of experiments at low- and highly enriched uranium reactors.
