  A large fraction of massive stars evolve in interacting binary systems, which
dramatically modifies the outcome of stellar evolution. We investigated the
properties of blue supergiants in binary systems and whether they are suitable
for extragalactic distance determinations using the flux-weighted gravity
luminosity relationship (FGLR). This is a relationship between the absolute
bolometric magnitude $M_{\rm bol}$ and the spectroscopically determined
flux-weighted gravity $g_{\rm F} = g/T^4_{\rm eff}$. We computed a grid of
binary stellar evolution models with MESA and use the v2.1 BPASS models to
examine whether they are compatible with the relatively small scatter shown by
the observed relationship. Our models have initial primary masses of 9 - 30
$M_\odot$, initial orbital periods of 10 - 2511 days, mass ratio $q$ = 0.9, and
metallicity $Z$ = 0.02. We find that the majority of primary stars that produce
blue supergiant stages are consistent with the observed FGLR, with a small
offset towards brighter bolometric magnitudes. In between 1\% - 24\% of cases,
binary evolution may produce blue supergiants after a mass transfer episode,
that lie below the observed FGLR. A small number of such stars have been found
in extragalactic FGLR studies, suggesting that they may have evolved through
binary interaction. Some models with shorter periods could resemble blue
hypergiants and LBVs. We used CMFGEN radiative transfer models to investigate
the effects of unresolved secondaries on diagnostics for $T_{\rm eff}$ and $g$,
and the biases on the determination of interstellar reddening and $M_{\rm
bol}$. We find that the effects are small and within the observed scatter, but
could lead to a small overestimate of the luminosity, of $T_{\rm eff}$ and of
$g$ for extreme cases. We conclude that the observed FGLR can, in principle, be
well reproduced by close binary evolution models.
