Unfortunately Civ4 Complete (And BtS 3.13) have a different 1.74 patch than Vanilla 1.74 - so they are not mutually compatible.
You should be able to make this compatible by downloading
this file.
This is a zip archive that contains the CvGameCoreDLL.dll for Vanilla (in the Assets folder of the Zip-Archive) and all the files that differ between Warlords 2.13 and BtS3.13/2.13. For your Vanilla games you only need the one file in the Assets folder of the zip archive.
Now on the machine with Vanilla (not Complete) installed open
...\Firaxis Games\Sid Meier's Civilization 4\Assets\
rename the CvGameCoreDLL.dll that is located there to CvGameCoreDLL_174.dll (or whatever you like, just rename it
).
Move the CvGameCoreDLL.dll from the Assets folder of the Zip Archive (NOT the Warlords one) to ...\Firaxis Games\Sid Meier's Civilization 4\Assets\
Start a game, try MP - on normal Vanilla + BtS vs Vanilla - BtS this will work, on Vanilla vs Complete it should work, but there is no guarantee
And Welcome back to CFC