Magic-angle helical trilayer graphene (HTG), a heterostructure consisting of three graphene layers with equal consecutive twists, relaxes into an array of commensurate domains with narrow and topological moir’e bands. We perform a detailed analysis of the interacting phase diagram at integer fillings, combining analytical strong-coupling analysis and Hartree-Fock numerics. We uncover a rich phase diagram of multiple topological symmetry-broken Chern insulators with |C| as large as 6. In contrast to other magic-angle graphene systems, we find that HTG remains in the “strong-coupling” regime, in which interactions dominate over kinetic energy, at all fillings. For experimentally-accessible displacement fields, these states undergo continuous topological phase transitions to more complex orders, including charge density waves and Kekul’e-ordered phases. Finally, we discuss connections to recent experimental studies in HTG. Our results demonstrate the robust capability of HTG to host gate-tunable topological and symmetry-broken correlated phases.