  We study the dimer model on subgraphs of the square lattice in which vertices
on a prescribed part of the boundary (the free boundary) are possibly
unmatched. Each such unmatched vertex is called a monomer and contributes a
fixed multiplicative weight $z>0$ to the total weight of the configuration. A
bijection described by Giuliani, Jauslin and Lieb relates this model to a
standard dimer model but on a non-bipartite graph. The Kasteleyn matrix of this
dimer model describes a walk with transition weights that are negative along
the free boundary. Yet under certain assumptions, which are in particular
satisfied in the infinite volume limit in the upper half-plane, we prove an
effective, true random walk representation for the inverse Kasteleyn matrix. In
this case we further show that, independently of the value of $z>0$, the
scaling limit of the height function is the Gaussian free field with Neumann
(or free) boundary conditions, thereby answering a question of Giuliani et al.
