We present the $\tau$-invariant balanced quasi-shuffle algebra $\mathcal{ G}^{ \operatorname{ f}}$, whose elements formalize (combinatorial) multiple Eisenstein series as well as multiple q-zeta values. In particular, $\mathcal{ G}^{ \operatorname{ f}}$ has natural maps into these two algebras, and we expect these maps to be isomorphisms. Racinet studied the algebra $\mathcal{ Z}^f$ of formal multiple zeta values by examining the corresponding affine scheme DM. Similarly, we present the affine scheme BM corresponding to the algebra $\mathcal{ G}^{ \operatorname{ f}}$. We show that Racinet's affine scheme DM embeds into our affine scheme BM. This leads to a projection from the algebra $\mathcal{ G}^{ \operatorname{ f}}$ onto $\mathcal{ Z}^f$. Via the above natural maps, this projection corresponds to extracting the constant terms of multiple Eisenstein series or the limit $q\to1$ of multiple q-zeta values.
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