The interplay between interactions and topology in quantum materials is of extensive current interest. For the gapless case, whether and how electron correlations can drive topological states that are not adiabatically connected to a noninteracting system is an open and pressing problem. Here we study a periodic Anderson model in several lattices, with strongly correlated electrons coupled to two channels of conduction electrons. Strong correlations are demonstrated to cooperate with lattice symmetry to produce topological phases that have no Landau quasiparticles and show strange-metal behavior. We characterize the electronic topology in terms of surface states and valley and spin Hall conductivities, and identify candidate materials to realize the proposed phases. Our work opens a door to a variety of gapless topological phases without free-electron counterpart in a broad range of strongly correlated metals.