Is this EdDSA modification secure?
In your signature each signer only seems to add their secret $a$ in once. An attacker who could get someone to sign first $M_1$, then $M_2$ and finally both as a set could find $a = S_1 + S_2 - S_{12}$ $mod$ $l$, i.e. all but a few bits of the secret $a$, the rest being within easy brute force search.

Not sure if that's your only problem.