since the shift looks similar, then I suspect that there is a datum difference between the two files... examine those with their respective coordinates
What is the coordinate system of the basemap?
What is the coordinate system of the X, Y points?
You claim the latter is 'true' ... how do you know? In order to make that judgement, you would need to know the coordinate system... Are the data in decimal degrees? what is the datum? for example is it a GCS WGS84 (a geographic coordinate system with a WGS84 datuem) like you might get from a GPS unit?
More details would be useful.
You will have to provide an image of the displacement and/or values showing the shift. 'somewhere else' doesn't help much in resolving the issue. There is something wrong that is causing the displacement. It has been known that reported coordinates being in one system have indeed been in another because of an incorrect definition at some stage prior to you acquiring the data and using it.
so as indicated before, there are three possible reasons. First, is the datum shift... if the pins are supposed to be where the end of the error tip is, then is could be a datum shift ... but I don't know what the distance values are so I have no idea whether that is 100m or what. Second, since the differences are similar but not exact this could mean that the coordinates are close to what they are supposed to be, but the numeric precision is insufficient to get a 'true value'. This gives rise to the last possibility that I can think of... your coordinates are perfect but the image is slightly displaced with some distortion and a slight rotation.
So from those (albeit, not a complete list), can you rule out the ones that seem the least plausible and report any supplementary information you may not have thought relevant at the time of reporting the issue