  Motivated by exploring doped multi-orbital antiferromagnets (AFMs) and
altermagnets (ALMs) we explore minimal $t$-$J$ models on the square-octagon
lattice which favor such collinear magnetic orders in the regime where spin
exchange dominates. While the AFM order breaks translational and time-reversal
symmetries, the ALM state (equivalently, a `$d$-wave ferromagnet') features
multipolar order which separately breaks time-reversal and crystal rotation
symmetries but preserves their product leading to spin-split bands with zero
net magnetization. We study the mean field phase diagram of these models as we
vary doping and interactions, discovering regimes of weak and strong ALM order,
superconductivity including uniform $s$-wave and $d$-wave pairing states,
incipient $d$-wave pair density wave order, and phases with coexisting
singlet-triplet pairing and AFM/ALM orders which appear unstable to phase
separation and could host stripe order with longer-range interactions. Our
results may be relevant to doping or pressure studies of multiorbital ALM
materials.
