  We construct a family of $q$-series with rational coefficients satisfying a
variant of the extended double shuffle equations, which are a lift of a given
$\mathbb{Q}$-valued solution of the extended double shuffle equations. These
$q$-series will be called combinatorial (bi-)multiple Eisenstein series, and in
depth one they are given by Eisenstein series. The combinatorial multiple
Eisenstein series can be seen as an interpolation between the given
$\mathbb{Q}$-valued solution of the extended double shuffle equations (as
$q\rightarrow 0$) and multiple zeta values (as $q\rightarrow 1$). In
particular, they are $q$-analogues of multiple zeta values closely related to
modular forms. Their definition is inspired by the Fourier expansion of
multiple Eisenstein series introduced by Gangl-Kaneko-Zagier. Our explicit
construction is done on the level of their generating series, which we show to
be a so-called symmetril and swap invariant bimould.
