Higherature superconducting (HTS) cuprate materials, with the ability to carry large electrical currents with no resistance at easily reachable temperatures, have stimulated enormous scientific and industrial interest since their discovery in the 1980's. However, technological applications of these promising compounds have been limited by their chemical and microstructural complexity and the challenging processing strategies required for the exploitation of their extraordinary properties. The lack of theoretical understanding of the mechanism for superconductivity in these HTS materials has also hindered the search for new superconducting systems with enhanced performance. The unexpected discovery in 2008 of HTS iron-based compounds has provided an entirely new family of materials for studying the crucial interplay between superconductivity and magnetism in unconventional superconductors. Alkali-metal-doped iron selenide (A_xFe_2-ySe₂, A = alkali metal) compounds are of particular interest owing to the coexistence of superconductivity at relatively high temperatures with antiferromagnetism. Intrinsic phase separation on the mesoscopic scale is also known to occur in what were intended to be single crystals of these compounds, making it difficult to interpret bulk property measurements. Here, we use a combination of two advanced microscopy techniques to provide direct evidence of the magnetic properties of the individual phases. First, x-ray linear dichroism studies in a photoelectron emission microscope, and supporting multiplet calculations, indicate that the matrix (majority) phase is antiferromagnetic whereas the minority phase is nonmagnetic at room temperature. Second, cryogenic magnetic force microscopy demonstrates unambiguously that superconductivity occurs only in the minority phase. The correlation of these findings with previous microstructural studies and bulk measurements paves the way for understanding the intriguing electronic and magnetic properties of these compounds.