WordPress would interpret this as a different page/post, so I initially though that it would not be possible to achieve this, however I then had the idea of using a redirect.
I tried three different 301 redirect plugins without success, but then found that this can be achieved with the use of dummy pages and Mark Jaquith’s Page Links To plugin.
After installing Page Links to, set up a page with the slug https://www.mysite.com/patients/amenities by making a child page of the patients page called amenities.
Then scroll down to the Page Links to metabox of the page editor, select the option to link to A custom URL and enter the address https://www.mysite.com/patients/?target=amenities
This is a fairly convoluted process in that it requires a dummy page for each tab you want to link to, but it does work on my test site.
