i think i know what is going on here. assigning the new coordinate 'control' quality should do it, but you likely also want to do the following:
unfortunately when you bring in your data from the data collector, the data collector has assigned 'coordinates' for each point. lets say that you take an average rtk observation or double a total station observation to a point, then the resulting average is a coordinate in the data collector, all points get these, even imported points. our workflow is to erase all these data collector coordinates. select all/then advanced select/drop down to coordinate/then expand your data collector file and they should still all be highlighted. typically you can delete all these, unless you need some of them for something(if they are known good coordinates). then create a new coordinate for you base station and call it control quality, and typically adjust a network for the project.
in theory you do not have to delete all the coordinates that i am talking about and you could just select your base point, on the property page on the upper ribbon there is a point symbol, press the point symbol to assign the new coordinate to the point, and use the drop downs next to the coordinates to assign control quality. but if you skip the first step then it will shift and every point will have a flag because tbc will flag the difference between the data collector coordinate and the shifted observation coordinates.
you will also need to press the little red circle (upper or lower ribbon) to compute the project, but i think it does this on export as well.