  We investigate the propagation characteristics of two counter propagating
dust acoustic solitary waves (DASWs) undergoing a head-on collision, in the
presence of strong coupling between micron sized charged dust particles in a
complex plasma. A coupled set of nonlinear dynamical equations describing the
evolution of the two DASWs using the extended Poincar\'e-Lighthill-Kuo
perturbation technique is derived. The nature and extent of post collision
phase-shifts of these solitary waves are studied over a wide range of dusty
plasma parameters in a strongly and a weakly coupled medium. We find a
significant change in the nature and amount of phase delay in the strongly
coupled regime as compared to a weakly coupled regime. The phase shift is seen
to change its sign beyond a threshold value of compressibility of the medium
for a given set of dusty plasma parameters.
