There is a great resource of historic ships that lets you search by ship type, location, or name. It’s from Historic Naval Ships Association and in it I found a fleet tug that participated in operation Mulberry.
The ship had an interesting history during the operation:
Built to serve during WW II, USAT LT-5 moved military cargo under the Army Transportation Corps. She served in both the Atlantic and Pacific. On February 3, 1944, she sailed for Great Britain to assist in the preparations for Operation Overlord. LT-5 arrived off the Normandy coast on June 7 as part of Operation Mulberry. On June 8th while moored to a sunken LST, LT-5 was subjected to air attacks. Her log book for June 9 records that at 20:30 hours, “planes overhead. Everyone shooting at them. Starboard gunner got an F.W.” (German Luftwaffe fighter, the Focke Wulf.)
You can read the full entry here (Link).
LT-5 is located at the H Lee White Marine Museum in Oswego, NY (Link)