We study a minimal square-lattice model to examine the relation between momentum-space altermagnetic band geometry and real-space Ising criticality in a coupled fermion-Ising setting. In this model, itinerant fermions with explicitly spin-dependent nearest-neighbor hopping are locally coupled to transverse-field Ising moments. The hopping anisotropy parameter $\alpha$ tunes a $d_{x^2-y^2}$-type altermagnetic band geometry in momentum space, whereas the transverse field $h$ controls real-space antiferromagnetic (AFM) fluctuations and drives the finite-temperature AFM–paramagnetic (PM) transition of the Ising sector. Using determinant quantum Monte Carlo simulations, we show for the representative $\alpha <1$ case studied in detail, $\alpha =0.5$, that the momentum-space band-geometry fingerprint remains visible in the coupled system: the spin-resolved occupation imbalance $n_{\uparrow }\left(\mathbf{k}\right)-n_{\downarrow }\left(\mathbf{k}\right)$ displays a sign-changing structure that follows the underlying spin-split Fermi-surface geometry. By contrast, finite-size scaling of the staggered Ising order parameter indicates that, in the representative comparison between $\alpha =0.5$ and $1.0$, the extracted thermal critical fields remain numerically close on the scale resolved by our finite-size analysis at both $T/t=0.5$ and $0.2$. We argue that this separation reflects the coexistence of a tunable momentum-space altermagnetic band geometry with the nesting structure of the half-filled nearest-neighbor bare kinetic term, which remains an important constraint on the AFM channel in this benchmark. Our results show that, in this controlled setting, a visible altermagnetic band-geometry fingerprint in momentum space need not imply a comparably large shift of the collective real-space magnetic phase boundary.

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• Altermagnetism
• Phase transitions
• Strongly correlated systems
• Quantum Monte Carlo