Quasipatterns with an eightfold rotational symmetry and irregular two- and three-mode patterns are found in an experiment on optical pattern formation. The patterns exist in the transverse cross section of a laser beam that traverses a system built from a sodium vapor cell and a plane feedback mirror with a quarter-wave plate placed into the feedback loop. The occurrence of the quasipatterns is reproduced by numerical simulations and explained by amplitude equations that contain only odd-order terms and are derived from the microscopic model. The selection process is governed by the angle dependence of the cubic cross coupling coefficients in the amplitude equations. Up to our knowledge it provides the first experimental example for a stabilization mechanism proposed earlier that is based on an oscillatory dependence of the cubic cross coupling coefficient on the angle between the interacting wave vectors. The relationship to the prediction of quasipatterns in a similar setup without additional wave plate [Phys. Rev. A 53, 1072 (1996)] is discussed.