I'm not aware of a native way to easily do this generally.
you can create a custom field for the records you want this to work for, and have a UE that sets the custom field whenever the external id is changed to match it, and then have that custom field be included in global search....
this is fairly simple but does require a small script, but isn't great if you need it for all record types... since you'll have to have that script deployed on all record types too in order to set the customfield. you'd also need a custom field for transactions, entities, items, and one for any/all custom records... but if this is just to support a single record type this is probably the easiest way.
A more general solution might be to create some kind of suitelet form that gives them some options for record selection, and then a blank field for them to type the search string and do the search yourself for the external id and then display the results... this is NOT the global search though its effectively its own external id search page which isn't what you asked for but its a cleaner general solution in that you dont have a UEs everywhere setting custom fields to just hold the value of the external id....
there might be better ways to do this but those are the two approaches that come to my mind.