  We studied the multiband properties of two ultraluminous X-ray sources (2CXO
J225728.9-410211 = X-1 and 2CXO J225724.7-410343 = X-2) and their surroundings,
in the spiral galaxy NGC 7424. Both sources have approached X-ray luminosities
L_{X} ~ 10^{40} erg/s at some epochs. Thanks to a more accurate astrometric
solution (based on Australia Telescope Compact Array and Gaia data), we
identified the point-like optical counterpart of X-1, which looks like an
isolated B8 supergiant (M ~ 9 Msun, age ~ 30 Myr). Instead, X-2 is in a
star-forming region (size of about 100 pc x 150 pc), near young clusters and
ionized gas. Very Large Telescope long-slit spectra show a spatially extended
region of HeII 4686 emission around the X-ray position, displaced by about 50
pc from the brightest star cluster, which corresponds to the peak of
lower-ionization line emission. We interpret the HeII 4686 emission as a
signature of X-ray photo-ionization from the ULX, while the other optical lines
are consistent with UV ionization in an ordinary HeII region. The luminosity of
this He^{++} nebula puts it in the same class as other classical photo-ionized
ULX nebulae such as those around Holmberg II X-1 and NGC 5408 X-1. We locate a
strong (5.5-GHz luminosity nu L_{nu} ~ 10^{35} erg/s), steep-spectrum,
unresolved radio source at the peak of the low-ionization lines, and discuss
alternative physical scenarios for the radio emission. Finally, we use WISE
data to obtain an independent estimate of the reddening of the star-forming
clump around X-2.
