  We consider the $N\times N$ Hermitian matrix model with measure
$d\mu_{E,\lambda}(M)=\frac{1}{Z} \exp(-\frac{\lambda N}{4} \mathrm{tr}(M^4))
d\mu_{E,0}(M)$, where $d\mu_{E,0}$ is the Gaussian measure with covariance
$\langle M_{kl}M_{mn}\rangle=\frac{\delta_{kn}\delta_{lm}}{N(E_k+E_l)}$ for
given $E_1,...,E_N>0$. It was previously understood that this setting gives
rise to two ramified coverings $x,y$ of the Riemann sphere strongly tied by
$y(z)=-x(-z)$ and a family $\omega^{(g)}_{n}$ of meromorphic differentials
conjectured to obey blobbed topological recursion due to Borot and Shadrin. We
develop a new approach to this problem via a system of six meromorphic
functions which satisfy extended loop equations. Two of these functions are
symmetric in the preimages of $x$ and can be determined from their consistency
relations. An expansion at $\infty$ gives global linear and quadratic loop
equations for the $\omega^{(g)}_{n}$. These global equations provide the
$\omega^{(g)}_{n}$ not only in the vicinity of the ramification points of $x$
but also in the vicinity of all other poles located at opposite diagonals
$z_i+z_j=0$ and at $z_i=0$. We deduce a recursion kernel representation valid
at least for $g\leq 1$.
