Real Chern insulators in two-dimensional altermagnetic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Fe</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:msub> <mml:mi mathvariant="normal">S</mml:mi> <mml:mn>2</mm

Altermagnets, recently identified as a third class of collinear magnetic materials, have attracted significant attention in condensed matter physics. Despite this growing interest, the realization of real Chern insulators in intrinsic altermagnetic systems has rarely been reported. In this work, based on first-principles calculations and theoretical analysis, we identify monolayer ${\mathrm{Fe}}_{2}{\mathrm{S}}_{2}\mathrm{O}$ and ${\mathrm{Fe}}_{2}{\mathrm{Se}}_{2}\mathrm{O}$ as a novel class of two-dimensional altermagnetic real Chern insulators. We demonstrate that these materials possess altermagnetic ground states and host a nontrivial mirror real Chern number, leading to the emergence of symmetry-protected zero-dimensional corner states. Notably, these corner modes are spin-polarized, giving rise to a unique spin--corner coupling effect. We further show that the real Chern insulating phases and their associated corner states remain robust against spin--orbit coupling, as well as under both uniaxial and biaxial strain. Additionally, these materials exhibit pronounced linear dichroism and strong optical absorption. Our findings uncover the novel topological character of ${\mathrm{Fe}}_{2}{\mathrm{S}}_{2}\mathrm{O}$ and ${\mathrm{Fe}}_{2}{\mathrm{Se}}_{2}\mathrm{O}$, establishing them as promising platforms for exploring real Chern insulators in altermagnetic systems.