  We introduce "geometric" partial comodules over coalgebras in monoidal
categories, as an alternative notion to the notion of partial action and
coaction of a Hopf algebra introduced by Caenepeel and Janssen. The name is
motivated by the fact that our new notion suits better if one wants to describe
phenomena of partial actions in algebraic geometry. Under mild conditions, the
category of geometric partial comodules is shown to be complete and cocomplete
and the category of partial comodules over a Hopf algebra is lax monoidal. We
develop a Hopf-Galois theory for geometric partial coactions to illustrate that
our new notion might be a useful additional tool in Hopf algebra theory.
