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I'm not certain, but this may be related to a datum shift. The ArcGIS Online Basemaps are all in WGS_1984_Web_Mercator_Auxiliary_Sphere, but perhaps your shapefile is in NAD_1983 or NAD_1927. If so, you can use the Project Tool to project your shapefile into WGS_1984_Web_Mercator_Auxiliary_Sphere. You will need to use KB Article 21327 to download a word document that will help you determine which Geographic Transformation to choose. Otherwise, you can also perform the transformation in the Data Frame Properties under the Coordinate System tab. You will again need to use the above KB Article to determine what transformation you will need to use. The important things to figure out for this process are what are the two different datums and where are the data supposed to be on the Earth.
I had a look at the data superimposed on some basemap imagery in various projections and using different assumptions about the original data coordinates and there seems to be variability in the fit west to east almost as if it was compiled from different data sets: some in NAD27 and others NAD83 or WGS-84. On the west coast the data fit reasonably well as is, but as you go east they shift and look like they were NAD 27 coordinates or even NAD83 projected to NAD 27 a couple of times. Some places they are in between what you would get using the WGS_1984_to_NAD_1927_18 transformation.
Hard to judge the data in places outside the cities where the boundary lines are generalized on landforms and many places in the road networks there are ambiguities where there are few distinctive road shapes.
However, I'd be concerned about how the dataset was put together. It appears to have been compiled in geographically variable geographic coordinate systems including NAD 27 and NAD 83 and maybe even some incorrect transformations.
Hope there is a simpler explanation...
some specific examples:
In eastern Tijuana the WGS-84 defined electoral boundary shapefile fits well with street patterns, but in western Tijuana it does not and appears shifted to the west.
What appears to have happened to the western Tijuana data is that it was collected in WGS-84, but mistakenly redefined as NAD27 and then projected to WGS-84 perhaps along with data from eastern Tijuana that was truly in NAD27 coordinates and correctly projected.
Likely this has happened in other parts as well. To correct a specific sector of the data you would clip it out and project to NAD27 then simply redefine the data as WGS-84 in the properties for the shapefile in ArcCatalog.
Beware however that while there seems to be a straightforward transformation mistake between NAD 27 and WGS 84 in the Tijuana data divided along some shifting line, elsewhere it could have been more political.