I think the answer is best provided by 2 measures:
1) If this is Theseus' Ship... at any point that Theseus is present as owner, the description of any variant is correctly described. Functionally, this does allow for multiple ships, although realistically, only one ship of a single identity, prior to his last visit/command of "the" ship
2) whenever the object (i.e. the ship) retains A) a majority of parts comprising the state being measured (i.e. is it a ship) or B) if the original authority of state (i.e. owner/Theseus) documents a set of components comprising the minimum viable set representing an original unit (e.g. the main mast) of the composable object, ...as of the last moment that condition 1 was measured, it is still the object being measured. Whenever applicable criteria (A or B) cease to be fulfilled, the item becomes "remnants of <the object>". Note that you can have multiple sets of remnants, or combinations of new materials and remnants, but only one set will fulfill criteria (A or B) at a time.
In the case of humans, consciousness is the minimum viable set.
I think the thought experiment is flawed in that it assumes you can replace a majority of a thing, and also not meet the <presence of Theseus> requirement, and still refer to it as the same reference.