Although Alpha Phi Omega doesn't have any limitations on cross-membership, I do periodically see it caveatted with "the other group may have a problem". I've just never see any such group.
I know the NPC (hist White Sororities) have strict and exacting rules on internal cross-membership and pretty sure that NIC (HWF) and NPHC (HBF&S) have them as well. Theoretically you could cross groups (say AKA sorority in the NPHC and Delta Delta Delta in the NPC, but I'd want to know the rules in exacting detail first.)
I think there are some cross-membership limitations in the Music Professional/Honoraries like Delta Omicron, but I'm not sure.
But the only Greek Letter Organizations that get grouped with APO normally are Gamma Sigma Sigma, Omega Phi Alpha and Intercollegiate Knights (all currently much smaller) and I know of a woman who is both an APO brother and a GSS Sister (and in Circle K

)
Now I do know of chapters that have unwritten rules regarding pledging both Alpha Phi Omega and a Social Fraternity/Sorority in the same semester, but that strikes me as efforts to save people's sanity, not a competition with the Social.