As Google promised, they are working on a solution to show the publishers URL instead of the AMP cache URL in Google’s search results after you click on an AMP page. Yesterday Google posted a demo of how it works and how you can try it yourself, as I also announced at Search Engine Land.
Anyway, the issue was/is that Google will link you to the google.com/amp URL instead of your publisher URL:

Google has fixed this through some modern web solutions and modern browsers – it only works on Chrome version 71 or greater and yes, mobile only.
Here is how it looks:

Here is how you can try it yourself:
Using a technology called Signed HTTP Exchanges cached AMP URLs are transformed to the publisher domain on any AMP Cache. It is currently only enabled in Chrome version 71 or greater. At the time of writing, this requires installing from the Chrome Beta channel but it will soon be released to all Chrome channels.
If you are not using a mobile device such as a smartphone or a tablet, enable mobile emulation on your desktop browser. AMP pages are only displayed to mobile devices. Next, visit https://g.co/webpackagedemo.
This will display a search box. From there, enter a query such as [learn amp by example] and click on “Learn AMP by Example” for the ampbyexample.com home page. There are only a small number of publishers that have implemented this feature so far, so you may want to try this specific query. If you’ve completed these steps correctly, you will see “https://ampbyexample.com” displayed in the browser’s URL bar.
Again, this only solves the problem on super modern browsers but hey, it is a step forward.
Forum discussion at Twitter.

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