The autocomplete form element drop down persist when it is either searched or clicked to appear for a select multiple type autocomplete form.  Clicking on the page doesn't do anything and the only way you can uncover the rest of the form elements by searching for something which doesn't exist.

Scenario

Clean latest 3.6 weekly version:

$version = 2018120305.07; // 20181203 = branching date YYYYMMDD - do not modify!
// RR = release increments - 00 in DEV branches.
// .XX = incremental changes.

$release = '3.6.5+ (Build: 20190816)'; // Human-friendly version name

Created a test course
Edited the test course and opened the tags section
Clicked on the manage standard tags link
In that page added a number of tags (letters ranging from a through p)
Reloaded the edit course page so these are picked up
clicked on the show dropdown for the tags autocomplete
selection covers the course form buttons and does not disappear preventing the form from being submitted/cancelled

Workaround is to search for something which doesn't exist which allows for the buttons to be uncovered.