  We search for runaway former companions of the progenitors of nearby Galactic
core-collapse supernova remnants (SNRs) in the Tycho-Gaia astrometric solution
(TGAS). We look for candidates for a sample of ten SNRs with distances less
than $2\;\mathrm{kpc}$, taking astrometry and $G$ magnitude from TGAS and $B,V$
magnitudes from the AAVSO Photometric All-Sky Survey (APASS). A simple method
of tracking back stars and finding the closest point to the SNR centre is shown
to have several failings when ranking candidates. In particular, it neglects
our expectation that massive stars preferentially have massive companions. We
evolve a grid of binary stars to exploit these covariances in the distribution
of runaway star properties in colour - magnitude - ejection velocity space. We
construct an analytic model which predicts the properties of a runaway star, in
which the model parameters are the properties of the progenitor binary and the
properties of the SNR. Using nested sampling we calculate the Bayesian evidence
for each candidate to be the runaway and simultaneously constrain the
properties of that runaway and of the SNR itself. We identify four likely
runaway companions of the Cygnus Loop, HB 21, S147 and the Monoceros Loop. HD
37424 has previously been suggested as the companion of S147, however the other
three stars are new candidates. The favoured companion of HB 21 is the Be star
BD+50 3188 whose emission-line features could be explained by pre-supernova
mass transfer from the primary. There is a small probability that the
$2\;\mathrm{M}_{\odot}$ candidate runaway TYC 2688-1556-1 associated with the
Cygnus Loop is a hypervelocity star. If the Monoceros Loop is related to the
on-going star formation in the Mon OB2 association, the progenitor of the
Monoceros Loop is required to be more massive than $40\;\mathrm{M}_{\odot}$
which is in tension with the posterior for HD 261393.
