In the field of collaborative robotics, the ability to communicate spatial information like planned trajectories and shared environment information is crucial. When no global position information is available (e.g., indoor or GPS-denied environments), agents must align their coordinate frames before shared spatial information can be properly expressed and interpreted. Coordinate frame alignment is particularly difficult when robots have no initial alignment and are affected by odometry drift. To this end, we develop a novel multiple hypothesis algorithm, called TCAFF, for aligning the coordinate frames of neighboring robots. TCAFF considers potential alignments from associating sparse open-set object maps and leverages temporal consistency to determine an initial alignment and correct for drift, all without any initial knowledge of neighboring robot poses. We demonstrate TCAFF being used for frame alignment in a collaborative object tracking application on a team of four robots tracking six pedestrians and show that TCAFF enables robots to achieve a tracking accuracy similar to that of a system with ground truth localization. The code and hardware dataset are available at https://github.com/mit-acl/tcaff.