Engineering magnetic orders in topological insulators is critical to the realization of topological quantum phenomena such as the axion insulator state and the quantum anomalous Hall insulator state. Here we establish MnBi$_6$Te$_{10}$ as a tunable topological material platform where ferromagnetism and antiferromagnetism can be selectively obtained. We conduct a comprehensive measurement of ferromagnetic MnBi$_6$Te$_{10}$ bulk crystals via laser-based angle-resolved photoemission spectroscopy, and compare the results with those from their antiferromagnetic counterparts. For ferromagnetic MnBi$_6$Te$_{10}$, we observe a magnetically driven broken-symmetry gap of 15 meV at the topological surface state on the MnBi$_2$Te$_4$ termination, which disappears when the temperature is raised above the Curie temperature. In contrast, antiferromagnetic MnBi$_6$Te$_{10}$ exhibits gapless topological surface states on all terminations. We consider disorder in the form of Mn migration from MnBi$_2$Te$_4$ layers to the neighboring Bi$_2$Te$_3$ layers as a possible driving force for the delicate ferromagnetism. Our spectroscopic study establishes MnBi$_6$Te$_{10}$ as the first bulk MnBi$_{2n}$Te$_{3n+1}$ compound to host tunable topological orders due to its highly variable electronic and magnetic structures.
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