Use firefox as render engine for Android webview
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Mozilla started working on GeckoView, which does exactly what you need. This blog post points to some sample code.
Here is the master bug, in case you want to follow progress: https://bugzilla.mozilla.org/show_bug.cgi?id=geckoview
Non-trivially difficult. Depending on how much consistency you need between your new WebView implementation and the existing implementation you could have a lot of work to do. There is an existing Firefox browser on Android so you could start with that (which would hopefully save you a lot of time) but at the very least you would need to check that the Firefox libraries exist and are accessible, you would probably need to bundle them with your app to get them to work (which would massively increase the size of your app) and then hooking up all the WebView apis would involve a lot of work.
The good part: there is an ‘official’ geckoview tutorial about how to integrate GeckoView, and a geckoview demo project on github:
The sad part: the official mozilla download site does not host the libraries anymore, and the latest available zips are from 2014:
At some moment there was a geckoview gradle artifact and a AAR, but the URLs are now dead.
Conclusion: apparently, GeckoView development has been halted, since URLs from the wiki page are broken, the library has been removed from the Mozilla continuous integration, and the bugs keep piling up with nobody to handle them. Hopefully I am wrong.