Context. Thanks to the high-precision photometry from space missions such as Kepler and TESS, tidal perturbations and tilting of pulsations have been detected in more than a dozen binary systems. However, only two of these were gravity-mode (g-mode) pulsators. Aims. We aim to detect tidally perturbed g modes in additional binary systems and characterise them observationally. Methods. We performed a custom data reduction of the available Kepler and TESS photometry of a well-studied, published sample of 35 binary systems with γ Doradus (γ Dor) pulsators. For each target, we modelled the binary signal using a sum of 100 sine waves with frequencies at orbital harmonics and measured significant pulsation frequencies in an iterative pre-whitening analysis of the residual light curve. Pulsations are labelled as tidally perturbed g modes if they are part of both period-spacing patterns and multiplets spaced by integer multiples of the orbital frequency. After visual inspection and confirmation, the properties of these targets and g modes were characterised. Results. We detect tidally perturbed g-mode pulsations for five short-period binaries that are circularised and (almost) synchronously rotating: KIC 3228863, KIC 3341457, KIC 4947528, KIC 9108579, and KIC 12785282. Tidally perturbed g modes that occur within the same star and have the same mode identification (k, m), are found to have near-identical relative amplitude and phase modulations, which are within their respective 1 − σ uncertainties and also identical for the Kepler and TESS photometric passbands. By contrast, pulsations with different mode identifications (k, m) are found to exhibit different modulations. Moreover, the observed amplitude and phase modulations are correlated, indicating that the binary tides primarily distort the g-mode amplitudes on the stellar surface. The phase modulations are then primarily a geometric effect of the integration of the stellar flux over the visible stellar surface. All selected binaries also exhibit signal that resembles rotational modulation in the Fourier domain. In the case of KIC 3228863, this is caused by the presence of the known tertiary component, and for the other systems we hypothesise that it is caused by temperature variations on the stellar surface. Alternatively, the signal can be made up of overstable convective modes in the stellar core or may belong to the non-pulsating companion. Conclusions. While g-mode pulsation periods are known to be a direct probe of the deep interior stellar structure, the binary tides that cause the pulsation modulations are dominant in the outer stellar layers. Hence, in the future, tidally perturbed g modes may allow us to carry out core-to-surface asteroseismic modelling of tidally distorted stars.