It's not the same in the US. TRI and TRE don't exist.
To conduct training is covered under FAR 61.167 IE. with an ATP you can instruct other pilots for an aircraft that you are rated on... both in the aircraft and simulator.
To conduct a checkride that would either be as a TCE (Training Center Evaluator) or as a "check airman" these are not added onto your licence but is an approval that is obtained via your company or training center.
Below is FAR 61.167
(a) A person who holds an airline transport pilot certificate is entitled to the same privileges as a person who
holds a commercial pilot certificate with an instrument rating.
(b) An airline transport pilot may instruct—
(1) Other pilots in air transportation service in aircraft of the category, class, and type, as applicable, for which the
airline transport pilot is rated and endorse the logbook or other training record of the person to whom training has
been given;
(2) In flight simulators, and flight training devices representing the aircraft referenced in paragraph (b)(1) of this
section, when instructing under the provisions of this section and endorse the logbook or other training record of
the person to whom training has been given;
(3) Only as provided in this section, except that an airline transport pilot who also holds a flight instructor
certificate can exercise the instructor privileges under subpart H of this part for which he or she is rated; and
(4) In an aircraft, only if the aircraft has functioning dual controls, when instructing under the provisions of this
section.
(c) Excluding briefings and debriefings, an airline transport pilot may not instruct in aircraft, flight simulators, and
flight training devices under this section—
(1) For more than 8 hours in any 24-consecutive-hour period; or
(2) For more than 36 hours in any 7-consecutive-day period.
(d) An airline transport pilot may not instruct in Category II or Category III operations unless he or she has been
trained and successfully tested under Category II or Category III operations, as applicable.