Steps to reproduce:

1. Tag (1) any host. Linux host for example
2. Tag (2) standard linux template
3. Add template to host

Result:
Trigger prototype contain full set of tags (1 and 2) in inherited tags.

Trigger contain only one tag (1) in inherited tags.

But in fact - tags exist because service changing status for set of problem tags  (1+2)

So looks like this is just visualization issue.

Expected:
Trigger should have full set of tags.

P.S. please add optional possibility to filter triggers by inherited tags!