  Very-faint X-ray binaries (VFXBs) are a sub-class of black holes and neutron
stars in binaries that appear to be accreting at a very low rate. In addition
to providing interesting constraints on poorly understood forms of accretion,
elucidating the nature of VFXBs is particularly interesting for binary
evolution and population modeling. Through near-infrared (nIR) spectroscopy, we
here investigate the nature of the bursting neutron star and VFXB 1RXH
J173523.7$-$354013 (J1735), which persistently accretes at an X-ray luminosity
of $L_X \sim 10^{34} - 10^{35}~L_{\odot}$. Our analysis shows that the nIR
emission is dominated by that of the companion star, which we find to be a late
G or early K-type giant, making this the second neutron star identified as a
VFXB found to have a giant companion. We discuss how several of the system
properties are difficult to reconcile with a wind-fed symbiotic X-ray binary.
We therefore also propose an alternative scenario wherein J1735 is a wide
binary system (supported by the discovery of a 7.5 d modulation in the nIR
light curves) with a quiescent luminosity of $L_X \sim 10^{34} -
10^{35}~L_{\odot}$, in which the donor star is overflowing its Roche lobe. This
raises the possibility that J1735 may, every century or more, exhibit very long
and very bright outbursts during which it reaches accretion rates around the
Eddington limit like the neutron star Z sources.
