In this work we present an experimental study of the phase characteristics of magnetostatic quasi-surface waves (MQSW) in a tangentially magnetized yttrium iron garnet (YIG) thin film bilayer. The phase of MQSWs was measured as a function of air gap (g) between two magnonic waveguides. It is shown that the energy exchange between two YIG layers is accompanied by relatively sharp π-radian phase jumps of MQSWs, at specific values of g. The effect was studied at large wavelengths of the MQSWs, and at a propagation distance of 1 cm. The theory predicts that the slope of the phase shift (∂φ/∂g) at submicrometric values of g can be of the order of a few degrees per nm.