Thermally-induced refractive index gratings (RIG) in high-power fiber laser systems lead to transverse mode instabilities (TMI) above a certain average power threshold. The effect of TMI is currently the main limitation for the further average power scaling of fiber lasers and amplifiers with nearly diffraction-limited beam quality. In this work we experimentally investigate, for the first time, the growth of the RIG strength by introducing a phase-shift between the RIG and the modal interference pattern in a fiber amplifier. The experiments reveal that the RIG is strong enough to couple energy between different transverse modes even at powers significantly below the TMI threshold, provided that the introduced phase-shift is high enough. This indicates that, as the strength of the RIG further increases with increasing average output power, the RIG becomes more and more sensitive to even small noise-induced phase-shifts, which ultimately trigger TMI. Furthermore, it is shown that a beam cleaning also occurs when a positive phase-shift is introduced, even above the TMI threshold. This finding will pave the way for the development of a new class of mitigation strategies for TMI, which key feature is the control of the introduced phase-shift.