We study a class of anomalies associated with time-reversal and spatial-reflection symmetry in (2+1)-dimensional bosonic topological phases of matter. In these systems, the topological quantum numbers of the quasiparticles, such as the fusion rules and braiding statistics, possess a Z(2) symmetry which can be associated with either time reversal (denoted Z(2)(T)) or spatial reflections. Under this symmetry, correlation functions of all Wilson loop operators in the low-energy topological quantum field theory (TQFT) are invariant. However, the theories that we study possess a severe anomaly associated with the failure to consistently localize the symmetry action to the quasiparticles, precluding even defining a consistent notion of symmetry fractionalization in such systems. We present simple sufficient conditions which determine when Z(2)(T) symmetry localization anomalies exist in general. We present an infinite series of TQFTs with such anomalies, some examples of which include USp(4)(2) Chern-Simons (CS) theory and SO(4)(4) CS theory. The theories that we find with these Z(2)(T) anomalies can all be obtained by gauging the unitary Z(2) subgroup of a different TQFT with a Z(2)(T) symmetry. We further show that the anomaly can be resolved in several distinct ways: (1) the true symmetry of the theory is Z(2)(T), or (2) the theory can be considered to be a theory of fermions, with T-2 = (-1)(Nf) corresponding to fermion parity. Finally, we demonstrate that theories with the Z(2)(T) localization anomaly can be compatible with Z(2)(T) if they are "pseudorealized" at the surface of a (3+1)D symmetry-enriched topological phase. The "pseudorealization" refers to the fact that the bulk (3+1)D system is described by a dynamical Z(2) gauge theory and thus only a subset of the quasiparticles are truly confined to the surface.