In their case the exterritoriality attaches to the vessel.
"Encyclopaedia Britannica, 11th Edition, Volume 10, Slice 1" by Various
Exterritoriality of foreign man-of-war, =4=, 122-25.
"The Life of John Marshall Volume 4 of 4" by Albert J. Beveridge
Thus, thirdly, foreign monarchs and envoys, foreign men-of-war, and foreign armed forces must be granted exterritoriality.
"International Law. A Treatise. Volume I (of 2)" by Lassa Francis Oppenheim