I am wondering if the construction of principal bundles associated to vector bundles can be generalized to linear fiber spaces in the category of complex/(real) analytic spaces. Linear fiber spaces allow the fiber rank to jump between different points and any linear fiber space is the spectrum of the symmetric algebra defined by a coherent sheaf.
When the base and the total space are smooth manifolds then one may of course construct the principal frame bundle and reductions of the structure group then tells one about potential additional structures on the original vector bundle.
Is there a similar construction for linear fiber spaces over analytic spaces? Is there perhaps a similar construction in the category of singular schemes?
I know that group spaces/schemes and principal fiber spaces/scheme have natural definitions in these categories. But I have not been able to find any mention of the above generalization.