  In metric-affine theories of gravity such as the C-theories, the spacetime
connection is associated to a metric that is nontrivially related to the
physical metric. In this article, such theories are rewritten in terms of a
single metric and it is shown that they can be recast as effectively nonlocal
gravity. With some assumptions, known ghost-free theories with non-singular and
cosmologically interesting properties may be recovered. Relations between
different formulations are analysed at both perturbative and nonperturbative
levels taking carefully into account subtleties with boundary conditions in the
presence of integral operators in the action, and equivalences between theories
related by nonlocal redefinitions of the fields are verified at the level of
equations of motion. This suggests a possible geometrical interpretation of
nonlocal gravity as an emergent property of non-Riemannian spacetime structure.
