$$\begin{array}{ccccc} \text{schemes} & \longrightarrow & \text{locally ringed spaces} & \longrightarrow & \text{ringed spaces} \\ | && | && | \\ \text{quasi-coherent sheaves} & \longrightarrow & \text{?} & \longrightarrow & \text{module sheaves}\end{array}$$
What do you suggest for $?$, fitting into this picture?
This is a soft question, but perhaps I can make it more precise: I would like to know if there is any reasonable substack $\mathsf{LMod}$ of $\mathsf{Mod} : \mathsf{LRS} \to \mathsf{SymMonCat}^{\mathrm{op}}$ such that $\mathsf{LMod}(X)$ preserves "much" of the structure of $X$ (in particular we cannot just use the forgetful functor $\mathsf{LRS} \to \mathsf{RS}$). I would like to see something different from quasi-coherent or coherent sheaves, which really used the local rings. For example, when $x \in \overline{\{y\}}$, one can require that the canonical homomorphism $M_x \otimes_{\mathcal{O}_{X,x}} \mathcal{O}_{X,y} \to M_y$ is an isomorphism modulo $(\mathfrak{m}_y)^n$, but this is a little bit weak.