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.
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