<p>Momentum-space nonsymmorphic symmetries, stemming from the projective algebra of synthetic gauge fields, can modify the manifold of the Brillouin zone and lead to a variety of topological phenomena. Recently, higher-order topological insulators (HOTIs) are predicted to be stabilized by these symmetries but call for experimental confirmation. We present an acoustic realization of HOTIs protected by a pair of anticommutative momentum-space glide reflections. We confirm the presence of momentum-space glide reflection from the measured momentum half translation of edge bands and their momentum-resolved probability distribution using a cylinder geometry made of acoustic resonator arrays. In particular, we observe both intrinsic and extrinsic HOTI features in such a cylinder: hopping strength variation along the open boundary leads to a bulk gap closure, while that along the closed boundary results in an edge gap closure. In addition, we confirm the presence of quadrupole corner modes with transmission and field distribution measurements. Our observation enriches the study of topological physics of momentum-space nonsymmorphic symmetries.</p>