  Galactic binaries with orbital periods less than $\approx$1 hr are strong
gravitational wave sources in the mHz regime, ideal for the Laser
Interferometer Space Antenna (LISA). In fact, theory predicts that \emph{LISA}
will resolve tens of thousands of Galactic binaries individually with a large
fraction being bright enough for electromagnetic observations. This opens up a
new window where we can study a statistical sample of compact Galactic binaries
in both, the electromagnetic as well the gravitational wavebands. Using
multi-messenger observations we can measure tidal effects in detached double WD
systems, which strongly impact the outcome of WD mergers. For accreting WDs as
well as NS binaries, multi-messenger observations give us the possibility to
study the angular momentum transport due to mass transfer. In this white paper
we present an overview of the opportunities for research on Galactic binaries
using multi-messenger observations and summarize some recommendations for the
2020 time-frame.
