Collisions between a 0.8 M. red giant and impactors of mass 0.4 and 0.6 M. are simulated using a three-dimensional smoothed particle hydrodynamics (SPH) code. The red giant is modeled as a massive core surrounded by a gaseous envelope, while the impactor, assumed to be a white dwarf or a main-sequence star, is treated as single particle. Encounters at various impact parameters and with relative velocities pertinent to globular clusters (10 km s-1) are studied. The effect of the encounter on the red giant depends strongly on the impact parameter. We find that if the impactor passes within twice the radius of the red giant (2R(RG)), mass loss occurs from the red giant envelope and both objects are left bound in elliptical orbits. Calculations suggest that if the impactor has an initial distance at closest approach of approximately 1.6R(RG) or less, the impactor will spiral in to the red giant envelope on subsequent periastron passages. For bound systems with a greater separation, the orbit may circularize with a radius approximately double that of the initial encounter. In both cases the system will undergo a common envelope phase: in the latter case the red giant will eventually fill its Roche lobe as it expands up the giant branch. In the case of a sufficiently small closest approach (R(min) < R(RG)), the impactor is immediately captured by the core of the red giant. Both objects form a binary system orbiting inside a common envelope formed from the original envelope of the giant. Acting like an "eggbeater," the system spins up the surrounding gas, leading to further mass loss while spiraling together. With a white drawf impactor, the likely outcome of the common envelope phase will be a tight white dwarf/white dwarf binary that will merge in less than a Hubble time. In the case of a main-sequence star impactor, a tight binary comprised of the main-sequence star and a white dwarf (the red giant core) is likely to be produced, with the main-sequence star filling its Roche lobe.