  We discovered eclipses and dips in two luminous (and highly variable) X-ray
sources in M\,51. One (CXOM51 J132943.3$+$471135) is an ultraluminous supersoft
source, with a thermal spectrum at a temperature of about 0.1 keV and
characteristic blackbody radius of about $10^4$ km. The other (CXOM51
J132946.1$+$471042) has a two-component spectrum with additional thermal-plasma
emission; it approached an X-ray luminosity of $10^{39}$erg s$^{-1}$ during
outbursts in 2005 and 2012. From the timing of three eclipses in a series of
{\it Chandra} observations, we determine the binary period ($52.75 \pm 0.63$
hr) and eclipse fraction ($22\% \pm 0.1\%$) of CXOM51 J132946.1$+$471042. We
also identify a blue optical counterpart in archival {\it Hubble Space
Telescope} images, consistent with a massive donor star (mass of $\sim$20--35
$M_{\odot}$). By combining the X-ray lightcurve parameters with the optical
constraints on the donor star, we show that the mass ratio in the system must
be $M_2/M_1 \gtrsim ~18$, and therefore the compact object is most likely a
neutron star (exceeding its Eddington limit in outburst). The general
significance of our result is that we illustrate one method (applicable to
high-inclination sources) of identifying luminous neutron star X-ray binaries,
in the absence of X-ray pulsations or phase-resolved optical spectroscopy.
Finally, we discuss the different X-ray spectral appearance expected from
super-Eddington neutron stars and black holes at high viewing angles.
