However I have recompiled the framework using the same Xcode and it still gives me this error.
I hit this issue with Xcode 10 beta, which adds Swift 4.2. My issue was unrelated to Carthage or any external dependencies.
Go to YourTarget > Build Settings > Swift Language Version for all of your Xcode targets.
Mine were all set to Swift 4.
After adding a Copy Files Build Phase, I started getting the error
:0: error: module compiled with Swift 4.1.2 cannot be imported by the Swift 4.2 compiler: /Users/user/Library/Developer/Xcode/DerivedData/MyApp-faskfrihumfcdibjxisjvbfshppp/Build/Intermediates.noindex/MyApp.build/Debug/MyAppExtension.build/Objects-normal/x86_64/MyClass~partial.swiftmodule
Command MergeSwiftModule failed with a nonzero exit code
I then set each target's Swift Language Version to Swift 4.2, which eliminated the error.
I'm not sure why the error was occuring in the first place, though, as having all targets set to Swift 4 should be equally as valid.