<p>We present a general set of functionals for which phase-space Feynman path integrals of Schrödinger-type on a torus have rigorous interpretations. For any functional in this set, the time-slicing approximation of the phase-space path integral converges uniformly with respect to the ending point of position paths and the starting point of momentum paths on any compact subset. The use of piecewise bicharacteristic paths yields natural convergence. Our set is closed under addition and multiplication, generating integrable functionals for phase-space path integrals. Moreover, some properties analogous to those of classical integrals hold for these phase-space path integrals. Specifically, the order of phase-space path integrals is interchangeable with time integrals or certain limits.</p>