  It has been observed that, given an algebraic quantum field theory (AQFT) on
a manifold $M$ and an open cover $\{M_\alpha\}$ of $M$, it is typically not
possible to recover the global algebra of observables on $M$ by simply gluing
the underlying local algebras subordinate to $\{M_\alpha\}$. Instead of gluing
local algebras, we introduce a gluing construction of AQFTs subordinate to
$\{M_\alpha\}$ and we show that for simple examples of AQFTs, constructed out
for geometric data, gluing the local AQFTs subordinate to $\{M_\alpha\}$
recovers the global AQFT on $M$. This result implies that the local AQFTs
contain more information than the local algebras.
