  When a main sequence star undergoes Roche lobe overflow onto a supermassive
black hole (SMBH) in a circular extreme mass ratio inspiral (EMRI), a phase of
steady mass transfer ensues. Over millions of years, the binary evolves to a
period minimum before reversing course and migrating outwards. Because the time
interval between consecutive EMRIs is comparable to the mass-transfer
timescale, the semi-major axes of two consecutive mass-transferring EMRIs will
cross on a radial scale < few AU. We show that such EMRI crossing events are
inevitably accompanied by a series of mildly relativistic, grazing physical
collisions between the stars. Each collision strips a small quantity of mass,
primarily from the more massive star, which generally increases their radial
separation to set up the next collision after a delay of decades to centuries
(or longer) set by further gravitational radiation. Depending on the mass of
the SMBH, this interaction can result in N ~ 1-1e4 gas production events of
mass Msun/N, thus powering a quasi-periodic sequence of SMBH accretion-powered
flares over a total duration of thousands of years or longer. Although the EMRI
rate is 2-3 orders of magnitude lower than the rate of tidal disruption events
(TDE), the ability of a single interacting EMRI pair to produce a large number
of luminous flares - and to make more judicious use of the available stellar
fuel - could make their observed rate competitive with the TDE rate, enabling
them to masquerade as "TDE Imposters." We predict flares with luminosities that
decay both as power laws shallower than t^(-5/3) or as decaying exponentials.
Viscous spreading of the gas disks produced by the accumulation of previous
mass-stripping events places substantial mass on radial scales > 10-100 AU,
providing a reprocessing source required to explain the unexpectedly high
optical luminosities of some flares.
