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Correct me if I'm wrong (I'm not a DO), but I believe that DOs who complete ACGME residencies are accredited through the NBME boards (the MD pathway) and not through the AOA/osteopathic boards.
The NBME boards actually do recognize AOA-accredited medical schools, which is the only reason why DOs can participate in the regular match.
The problem isn't the ACGME or the NBME. If an MD did hypothetically complete an AOA residency, then they would need to be accredited by the osteopathic boards that oversaw his/her postgraduate education.
The problem is that there is no reciprocity with the osteopathic boards - although the NBME recognizes the DO degree and will grand board certification to a DO who completed an ACGME residency, the osteopathic boards will not do the same for an MD.
That's why MDs cannot participate in the AOA match.
This has been discussed ad nauseam on SDN.
1) NBME administers the USMLE and has nothing to do with board certification for post-graduate training (MD or DO).
2) AOA doesn't allow MDs to participate in AOA residencies because MDs don't get OMM training in medical school which is required for ALL AOA internship years (any specialty). This is irrespective of how little or how extensive the OMM component of the internship is (1hr/month vs. 1hr/day). If you are not trained in OMM, you can not complete this requirement, even if it is only for 1hr during your entire training, and therefore you can not finish internship year. (I'm not arguing for or against this policy; I'm just stating facts)!
3) DOs who complete ACGME training do have the option to become board certified by the respective AOA specialty board in ADDITION to the ABMS (MD) board by proving training equivalence and taking the exam, if they choose to do so (and many do)...