  Given a domain Y in a complex manifold X, it is a difficult problem with no
general solution to determine whether Y has a schlicht envelope of holomorphy
in X, and if it does, to describe the envelope. The purpose of this paper is to
tackle the problem with the help of a smooth 1-dimensional foliation F of X
with no compact leaves. We call a domain Y in X an interval domain with respect
to F if Y intersects every leaf of F in a nonempty connected set. We show that
if X is Stein and if F satisfies a new property called quasiholomorphicity,
then every interval domain in X has a schlicht envelope of holomorphy, which is
also an interval domain. This result is a generalization and a global version
of a well-known lemma from the mid-1980s. We illustrate the notion of
quasiholomorphicity with sufficient conditions, examples, and counterexamples,
and present some applications, in particular to a little-studied boundary
regularity property of domains called local schlichtness.
