  The method of variational completion allows one to transform an (in
principle, arbitrary) system of partial differential equations -- based on an
intuitive ``educated guess'' -- into the Euler-Lagrange one attached to a
Lagrangian, by adding a canonical correction term. Here, we extend this
technique to theories that involve at least two sets of dynamical variables :
we show that an educated guess of the field equations with respect to one of
these sets of variables only is sufficient to variationally complete these
equations and recover a Lagrangian for the full theory, up to terms that do not
involve the respective variables.
  Applying this idea to natural metric-affine theories of gravity, we prove
that, starting from an educated guess of the metric equations only, one can
find the full metric equations, together with a generally covariant Lagrangian
for the theory, up to metric-independent terms; the latter terms (which can
only involve the distortion of the connection) are then completely classified
over 4-dimensional spacetimes, by techniques pertaining to differential
invariants.
