We present a set of second-order, time-reversible algorithms for the isothermal (NVT) molecular-dynamics (MD) simulation of systems with mixed hard-core/continuous potentials. The methods are generated by combining real-time Nose thermostats with our previously developed Collision Verlet algorithm [Mol. Phys. 98, 309 (1999)] for constant energy MD simulation. In all we present five methods, one based on the Nose-Hoover [Phys. Rev. A 31, 1695 (1985)] equations of motion and four based on the Nose-Poincare [J. Comput. Phys. 151, 114 (1999)] real-time formulation of Nose dynamics. The methods are tested using a system of hard spheres with attractive tails and all correctly reproduce a canonical distribution of instantaneous temperature. The Nose-Hoover based method and two of the Nose-Poincare methods are shown to have good energy conservation in long simulations.