  This article introduces and studies the tight approximation property, a
property of algebraic varieties defined over the function field of a complex or
real curve that refines the weak approximation property (and the known
cohomological obstructions to it) by incorporating an approximation condition
in the Euclidean topology. We prove that the tight approximation property is a
stable birational invariant, is compatible with fibrations, and satisfies
descent under torsors of linear algebraic groups. Its validity for a number of
rationally connected varieties follows. Some concrete consequences are: smooth
loops in the real locus of a smooth compactification of a real linear algebraic
group, or in a smooth cubic hypersurface of dimension at least 2, can be
approximated by rational algebraic curves; homogeneous spaces of linear
algebraic groups over the function field of a real curve satisfy weak
approximation.
