  We present a search for binary neutron star mergers that produced
gravitational-waves during the first observing run of Advanced LIGO and
gamma-ray emission seen by either \textit{Swift}-BAT or Fermi-GBM, similar to
GW170817 and GRB 170817A. We introduce a new method using a combined ranking
statistic to detect sources that do not produce significant gravitational-wave
or gamma-ray burst candidates individually. The current version of this search
can increase by 70% the detections of joint gravitational-wave and gamma-ray
signals. We find one possible candidate observed by LIGO and Fermi-GBM, 1-OGC
151030, at a false alarm rate of 1 in 13 years. If astrophysical, this
candidate would correspond to a merger at $187^{+99}_{-87}\,$Mpc with
source-frame chirp mass of $1.30^{+0.02}_{-0.03}\,\mathrm{M}_{\odot}$. If we
assume the viewing angle must be $<30^{\circ}$ to be observed by
\textit{Fermi}-GBM, our estimate of the distance would become
$224^{+88}_{-78}\,$Mpc. By comparing the rate of binary neutron star mergers to
our search-estimated rate of false alarms, we estimate that there is a 1 in 4
chance this candidate is astrophysical in origin.
