We theoretically study the effect of pressure fluctuations on the Richtmyer-Meshkov (RM) unstable interface in approximation of ideal incompressible immiscible fluids and two-dimensional flow. Pressure fluctuations are treated as an effective acceleration directed from the heavy to light fluid with inverse square time dependence. The group theory approach is applied to analyze large-scale coherent dynamics, solve the complete set of the governing equations, and find regular asymptotic solutions describing RM bubbles. A strong effect is found, for the first time to our knowledge, of pressure fluctuations on the interface morphology and dynamics. In the linear regime, a nearly flat bubble gets more curved, and its velocity increases for strong pressure fluctuations and decreases otherwise. In the nonlinear regime, solutions form a one-parameter family parameterized by the bubble front curvature. For the fastest stable solution in the family, the RM bubble is curved for strong pressure fluctuations and is flattened otherwise. The flow is characterized by the intense motion of the fluids in the vicinity of the interface, effectively no motion away from the interface, and presence of shear at the interface leading to formation of smaller scale vortical structures. Our theoretical results agree with and explain existing experiments and simulations and identify new qualitative and quantitative characteristics to evaluate the strength of pressure fluctuations in experiments and simulations.