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.
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