TYPO3 Core

In TYPO3 9.2 you introduced a new hook to extend search query in list view:
https://docs.typo3.org/c/typo3/cms-core/master/en-us/Changelog/9.2/Feature-71911-AddConstraintHookInDatabaseRecordListMakeSearchString.html

Because the LiveSearch is very similar implemented, it would be easily possible to add such constraint hook there as well.

The place for the hook could be \TYPO3\CMS\Backend\Search\LiveSearch\LiveSearch::makeQuerySearchByTable right before checking if $constraints is empty.

Without such hook, you can only register a XClass to extend LiveSearch.