  We propose a sheaf-theoretic approach to the theory of differential calculi
on quantum principal bundles over non-affine bases. After recalling the affine
case we define differential calculi on sheaves of comodule algebras as sheaves
of covariant bimodules together with a morphism of sheaves -- the differential
-- such that the Leibniz rule and surjectivity hold locally. The main class of
examples is given by covariant calculi over quantum flag manifolds, which we
provide via an explicit Ore extension construction. In a second step we
introduce principal covariant calculi by requiring a local compatibility of the
calculi on the total sheaf, base sheaf and the structure Hopf algebra in terms
of exact sequences. In this case Hopf--Galois extensions of algebras lift to
Hopf--Galois extensions of exterior algebras with compatible differentials. In
particular, the examples of principal (covariant) calculi on the quantum
principal bundles $\mathcal{O}_q(\mathrm{SL}_2(\mathbb{C}))$ and
$\mathcal{O}_q(\mathrm{GL}_2(\mathbb{C}))$ over the projective space
$\mathrm{P}^1(\mathbb{C})$ are discussed in detail.
