  We study the set of solutions $(\omega_{g,n})_{g \geq 0,n \geq 1}$ of
abstract loop equations. We prove that $\omega_{g,n}$ is determined by its
purely holomorphic part: this results in a decomposition that we call "blobbed
topological recursion". This is a generalization of the theory of the
topological recursion, in which the initial data $(\omega_{0,1},\omega_{0,2})$
is enriched by non-zero symmetric holomorphic forms in $n$ variables
$(\phi_{g,n})_{2g - 2 + n > 0}$. In particular, we establish for any solution
of abstract loop equations: (1) a graphical representation of $\omega_{g,n}$ in
terms of $\phi_{g,n}$; (2) a graphical representation of $\omega_{g,n}$ in
terms of intersection numbers on the moduli space of curves; (3) variational
formulae under infinitesimal transformation of $\phi_{g,n}$ ; (4) a definition
for the free energies $\omega_{g,0} = F_g$ respecting the variational formulae.
We discuss in detail the application to the multi-trace matrix model and
enumeration of stuffed maps.
