  We present a new dynamical scenario for the formation of short-period binary
millisecond pulsars in globular clusters. Our work is motivated by the recent
observations of 20 radio pulsars in 47 Tuc. In a dense cluster such as 47 Tuc,
most neutron stars acquire binary companions through exchange interactions with
primordial binaries. The resulting systems have semimajor axes in the range
\~0.1-1 AU and neutron star companion masses ~1-3 Msun. For many of these
systems we find that, when the companion evolves off the main sequence and
fills its Roche lobe, the subsequent mass transfer is dynamically unstable.
This leads to a common envelope phase and the formation of short-period neutron
star - white dwarf binaries. For a significant fraction of these binaries, the
decay of the orbit due to gravitational radiation will be followed by a period
of stable mass transfer driven by a combination of gravitational radiation and
tidal heating of the companion. The properties of the resulting short-period
binaries match well those of observed binary pulsars in 47 Tuc.
