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Repeating earthquake sequences have been actively investigated to clarify many aspects of earthquake physics. The two particularly well‐studied sequences, known as the Los Angeles and San Francisco repeaters, have several intriguing observations, including their long (for the seismic moment) recurrence times that would suggest stress drops of 300 MPa based on typical assumptions, near‐syncronized timing prior to 2004, and higher than typical inferred stress drops (of 25 to 65 MPa, up to 90 MPa locally), but not as high as the recurrence times suggest. Here we show that all these observations are self‐consistent, in the sense that they can be reproduced in a single fault model. The suitable models build on the standard rate‐and‐state fault models, with velocity‐weakening patches imbedded into a velocity‐strengthening region, by adding either enhanced dynamic weakening during seismic slip or elevated normal stress on the patches, or both, to allow for the higher stress drops. Such models are able to match the observed average properties of the San Francisco and Los Angeles repeaters, as well as the overall nontrivial scaling between the recurrence time and seismic moment exhibited by many repeating sequences as a whole, for reasonable parameter choices based on experiments and theoretical studies. These models are characterized by the occurrence of substantial and variable aseismic slip at the locations of the repeating sources, which explains their atypical relation between recurrence interval and seismic moment, induces variability in the repeating source properties as observed, and results in their neither slip‐ nor time‐predictable behavior. Key Points: Enhanced dynamic weakening and/or elevated normal stress are used to reproduce observationsHigh stress drops, interaction, and repeat times of SF/LA repeaters are simultaneously reproducedAseismic slip between repeating events is key to explaining their source properties and variability [ABSTRACT FROM AUTHOR]