Windows – gem not working in Cygwin

On a freshly installed Windows 7 Professional 64 Bit I installed Cygwin (64) and some of its packages, including Ruby. I have also installed Ruby using the Ruby installer, because I'll likely need it for both the default Windows shells and Cygwin.

Now when I try to execute a gem command like gem list or gem install foo, I get a weird error I failed to solve within the last few hours of searching the internet.

$ which ruby
/usr/bin/ruby

$ which gem
/usr/bin/gem

$ ruby -v
ruby 2.2.4p230 (2015-12-16 revision 53155) [x86_64-cygwin]

$ gem -v
2.4.8

$ gem list
ERROR:  Loading command: list (Fiddle::DLError)
        can't load kernel32
ERROR:  While executing gem ... (NoMethodError)
    undefined method `invoke_with_build_args' for nil:NilClass

$ gem install sass
ERROR:  Loading command: install (Fiddle::DLError)
        can't load kernel32
ERROR:  While executing gem ... (NoMethodError)
    undefined method `invoke_with_build_args' for nil:NilClass

However, with the native Windows version, from the Windows CMD, it works without any problems. I cannot however use native Windows Ruby stuff from Cygwin, because that gives me errors, but that's not the question here anyway.

With Process Monitor I figured out that Ruby tries to open C:\cygwin64\bin\kernel32.dll and fails, because that file is not there. I tried copying the kernel32.dll from C:\Windows\System32 and the one from C:\Windows\SysWOW64 into that Cygwin bin folder and still got the same error (apart from that it then said can't load kernel32.dll), although the Process Monitor didn't show a NAME NOT FOUND error anymore.

What magic is happening here? I'd really like to understand what's wrong here. I appreciate any help.