Apparently the problem is caused by not having clang modules enabled for the C targets.
To enable them go to your Xcode project and in all the C targets (the NIO ones start with a C like CAtomics), change the Enable Modules (C and Objective-C) to YES.
And add the following import paths:
-Xcc -fmodule-map-file=$(SRCROOT)/Protocol/Atem.xcodeproj/GeneratedModuleMap/CNIOAtomics/module.modulemap -Xcc -fmodule-map-file=$(SRCROOT)/Protocol/Atem.xcodeproj/GeneratedModuleMap/CNIOLinux/module.modulemap -Xcc Β -fmodule-map-file=$(SRCROOT)/Protocol/Atem.xcodeproj/GeneratedModuleMap/CNIODarwin/module.modulemap
You probably want to use an .xcconfig file to do that automatically.
swift package generate-xcodeproj --xcconfig-overrides path/to/your/.xcconfig
