Can I use Cygwin with Windows-native-only terminal apps?

I use a Cygwin on Windows relatively often. However – the Cygwin terminal app, mintty.exe, is… well, underwhelming. It has repaint issues, resizes don’t always get communicated like I would expect etc.

Then there are all sorts of terminal applications you can use on Windows, with examples such as, well, Windows Terminal, but also cmd.exe (which isn’t that great either), PuTTY and others.

I was wondering whether I can run a Cygwin shell somehow from within other popular Windows terminal apps.