  We systematically searched for gravity- and Rossby-mode period spacing
patterns in Kepler eclipsing binaries with $\gamma$ Doradus pulsators. These
stars provide an excellent opportunity to test the theory of tidal
synchronisation and angular momentum transport in F- and A-type stars. We
discovered 35 systems that show clear patterns, including the spectroscopic
binary KIC 10080943. Combined with 45 non-eclipsing binaries with $\gamma$ Dor
components that have been found using pulsation timing, we measured their
near-core rotation rates and asymptotic period spacings. We find that many
stars are tidally locked if the orbital periods are shorter than 10 days, in
which the near-core rotation periods given by the traditional approximation of
rotation (TAR) are consistent with the orbital period. Compared to the single
stars, $\gamma$ Dor stars in binaries tend to have slower near-core rotation
rates, likely a consequence of tidal spin-down. We also find three stars that
have extremely slow near-core rotation rates. To explain these, we hypothesise
that unstable tidally excited oscillations can transfer angular momentum from
the star to the orbit, and slow the star below synchronism, a process we refer
to as `inverse tides'.
