  Consider a $C_0$-semigroup $(e^{tA})_{t \ge 0}$ on a function space or, more
generally, on a Banach lattice $E$. We prove a sufficient criterion for the
operators $e^{tA}$ to be positive for all sufficiently large times $t$, while
the semigroup itself might not be positive. This complements recently
established criteria for the individual orbits of the semigroup to become
eventually positive for all positive initial values. We apply our main result
to study the qualitative behaviour of the solutions to various partial
differential equations.
