  We present a high-resolution echelle spectrum taken with the Very Large
Telescope and analyse archival Hubble Space Telescope photometry of the
recently identified companion of the eclipsing millisecond radio pulsar PSR
J1740-5340 in the globular cluster NGC 6397. From the spectrum, we show that
the companion is metal poor, as expected for a member of NGC 6397. Using
synthetic photometry, and assuming a true distance modulus of 12.13 +/- 0.15
mag and a colour excess of E(B-V) = 0.179, we derive a radius of 1.60 +/- 0.17
solar radii and an effective temperature of 5410 +/- 50 K, implying a
luminosity of 2.0 +/- 0.4 solar luminosities (1 sigma errors). These properties
make it similar to the so-called `sub-subgiants' in the old open cluster M67
and `red stragglers' in 47 Tuc, which have luminosities comparable to those of
turn-off stars, but cooler temperatures and larger radii. The light curve of
the companion is well described by ellipsoidal variations, and despite the
incomplete (~60%) phase coverage, we are able to derive good constraints on a
number of the system parameters. In particular, for the inclination we find a 2
sigma lower limit of i > 48 deg (i > 46 deg at 3 sigma). Assuming a pulsar mass
of 1.2 M_MSP < 2.4 solar masses, this implies a companion mass in the range
0.14 < M_comp < 0.38 solar masses. Combined with the photometric constraint, we
find a best fit for i ~ 50 deg and M_comp ~ 0.3 solar masses. We infer a Roche
lobe filling factor by radius of ~97%. [Abstract shortened]
