"To all staff and (security) agents we say: do not let this racist ship soil Tunisia's ports. Keep them out as our brothers in (the ports of) Zarzis and Sfax have already done," the UGTT union said in Facebook post.
The UGTT, the Tunisian General Labour Union, is best known as a joint winner of the 2015 Nobel Peace Prize.
Any plans the C-Star had to dock at Zarzis in southeastern Tunisia on Sunday were scuppered when local fishermen warned they would block access to a refuelling channel.
Chartered by extremist group Generation Identity and crewed by activists from France, Germany and Italy, the C-Star's declared aim is to disrupt the flow of migrant boats from north Africa to Europe.
They have dubbed their mission "Defend Europe" and say they want to expose what they see as a taxi service for illegal immigrants being operated by privately-funded boats helping to patrol waters off Libya.
Humanitarian organisations operating the rescue boats have branded Defend Europe a potentially dangerous stunt. Two of the NGO vessels were contacted by the C-Star when it passed through waters off Libya on Saturday.
"We start our operation off the Libyan coast and we advise you to leave the SAR (Search and Rescue) area because you're acting as a pull factor for human traffickers, making them millions," the crew was recorded as telling their counterparts on the Aquarius, a boat operated by SOS Mediterranee and Doctors without Borders.
"We will watch you and the days of your unwatched doings here are over."
A similar, apparently intimidatory, message was delivered to the crew of the Golfo Azzurro, operated by Spain's Pro-Active Open Arms, who responded by telling C-Star's staff to stop using a radio channel reserved for emergency communications.
If the C-Star cannot dock in Tunisia it is unclear where else it could go.
Tentative plans to refuel in either Greece or Sicily on the way down to Libya were scuppered by opposition to their mission in both places.
The mission has been dogged by setbacks since the boat was chartered in Djibouti in mid-May with funds raised by crowd-funding.
The boat was held up for a week in the Suez Canal by Egyptian authorities looking for weapons then delayed further in Cyprus, where some of the Sri Lankan crew got off and claimed refugee status, embarrassingly adding to the list of asylum-seekers the mission is supposed to help cut.
The mission did not respond to a request for comment from AFP.
