note game is on the verge of the next population milestone
note the base influence of 205
that's 105 (initial NON-building-related Base influence) plus two skyline towers, each of 50:
406
Skyline Tower
Base
50
and when the next population level is reached, see base influence lost the buildings' influence and dropped -2*50 to 105:
to test, i have modded some buildings to have base influence, and can confirm the amount lost corresponds to the exact sum of the influence from Base
there is no corresponding problem with e.g. InfluenceSource of Culture (tested same mod with culture instead of base, influence remained)
i assume -- but did not actually test -- Engineers and Investors are also ok (those 4 are only 4 types of InfluenceSource)
only the skyline tower has Base
finally, saving the game and then reloading it returns the influence to its proper value