Geometric phases of real wave functions in nonintegrable quantum billiards are measured using microwave resonators. They appear as a sign change of the wave function after a cyclic excursion around a diabolic point in the space of shapes of the resonator. For a path encircling a pointlike triple degeneracy of an integrable system, an unexpected sign change for two out of the three wave functions is found, which must be attributed to a hidden symmetry, and not to a Berry phase. PACS numbers: 41.20.Bt, 03.65.Bz, 84.40.Cb Solutions of the two dimensional wave equation with nonintegrable boundary conditions have found much interest recently in the context of quantum chaos. The systems under investigation are two dimensional quantum billiards for which classical mechanics is chaotic. Energy levels and wave functions of such systems can be determined by extensive numerical calculations or by analog experiments with microwave cavities [1]. In microwave experiments the analogy between the Schrodinger equation for a bound state and the electrodynamic wave equation is used.Most investigations have been carried out for fixed boundary conditions. However, for many quantal systems like vibrating molecules or nuclei, the boundary conditions are not fixed but oscillate themselves. These oscillations are often slow enough that they can be separated from the much faster oscillations of the bound states. In this case it is interesting to study the adiabatic evolution of wave functions as a function of external parameters describing the slow boundary oscillations. After a cyclic adiabatic evolution within the space of external parameters a quantum system will return to its initial state but may have picked up a geometric phase pz" in addition to the dynamical phase p&""= &' Jn E(t)dt.The general formalism to calculate geometric phases was developed by Berry [2]. For a spin coupled to a rotating magnetic field one finds ps"--mA, where m is the magnetic quantum number and 0 the solid angle enclosed by the rotating field vector. The existence of Berry phases in spin variables has been proven in experiments with neutrons [3], nuclei [4] (spin 2), and light[5] (spin 1). But a geometric phase can also be picked up by real valued wave functions evolving under changing boundary conditions. In molecules it gives rise to the molecular Aharonov-Bohm effect [6]. In the simplest case this will happen if a diabolic degeneracy is encircled by the path of the system in the space of external parameters [7]. Such diabolic points typically occur in nonintegrable real quantum systems where they can be enforced by varying two independent parameters [8]. At a diabolic point two energy surfaces drawn over the plane of the two external parameters just touch each other at one point forming a double cone (diabolo) [9]; see inset Upon a full rotation of B through 8 = 2vr (thus enclosing a solid angle of 0 = 27r) a sign change of the state occurs:signaling a Berry phase = +m. The sign change of a real wave function along a path around a diabolic point can be ...