Let $X$ be a scheme over a field $k$. There is a well-known
antiequivalence between locally free sheaves of
$\mathcal{O}_X$-modules and vector bundles over $X$. Given a module
$\mathcal F$ and a trivialisation $\{U_i\}$ with transition
functions given by matrices $f_{ij}$ with values in
$\mathcal{O}_{U_i \cap U_j}$ one takes $\mathbb{A}_{U_i}^n$ -s and
glues them according to transition functions given by the adjoint
matrices $f_{ij}^*$. This antiequivalence can also be described as
taking the relative spectrum of the sheafification of a presheaf $U
\mapsto Sym(\mathcal{F}(U))$.
More generally, given a coherent sheaf $\mathcal F$ and working locally, over an open $U$ one has $\mathcal{O}_X^n(U) \xrightarrow{f} \mathcal{O}_X^m(U) \to
\mathcal{F}(U) \to 0$ and one can construct a scheme $Y_U$ as follows: let $Y_U$ be the subscheme of
$\mathbb{A}_U^m$ defined by the ideal generated by $(\sum f_{1i}
x_i, \ldots, \sum f_{ni} x_i)$. Then one can glue $Y_U$-s together and get $Y$. The scheme $Y$ has a natural
structure of a linear space over $X$, i.e. there are maps $+: Y \times_X Y \to Y$, $\cdot: \mathbb{A}_k \times Y \to Y$ that satisfy the $k$-module
axioms. A morphism of linear spaces over $X$ is a map of schemes that
preserves these operations. To $Y$ one can associate the sheaf which
is the sheafification of the presheaf $U \mapsto Hom(Y|_U,
\mathbb{A}_k \times X)$, this defines an antiequivalence between the
category of coherent sheaves on $X$ and the category
of linear spaces over $X$.
The reduction functor is a functor from the category of schemes to the category of reduced schemes. In the latter category the product is the usual product of schemes followed by the application of reduction functor. Therefore, tautologically, $(-)_\mathrm{red}$ preserves products.
Assume $X$ reduced. In general, the linear space $X_\mathcal{F}$ associated to an $\mathcal{O}_X$-module $\mathcal F$ is not reduced. Since linear spaces are defined using diagrams (with products), and $(-)_\mathrm{red}$ is a product-preserving functor, $(X_\mathcal{F})_\mathrm{red}$ is a linear space and hence corresponds to some $\mathcal{O}_X$-module, call it $\mathcal{F}_\mathrm{red}$.
Is it possible to give a ``direct'' description the correspondence $\mathcal{F} \to \mathcal{F}^{red}$ without referring to the procedure described above?
Is it possble to characterise those sheaves of modules that correspond to reduced linear spaces?
