More than 4,000 are thought to have found their way into British ports in the last year, and the problem is becoming so serious that last week in London the UN's International Maritime Organisation (IMO) set up a working group to tackle the problem.
Most of the stowaways who make their way to Britain are economic refugees, anxious to take a slice of the West's wealth. Some are asylum seekers fleeing political upheaval. Others are in transit, hoping to continue their journey to the United States. But they all have one point in common: a desperate desire to get out.
To do so they are willing to take enormous risks, often squeezing themselves into small spaces, with minimal food and water and no light or heating.
Carrier bags are used to collect excrement and water bottles double as urine bottles, but the hygiene problems and discomfort are still enormous - especially if the stowaways are travelling in groups.
Three weeks ago, Dover port officials discovered one such group on board a lorry from France. Squashed into a 2ft 6in gap with little food and no sanitation, police found 31 Bangladeshis: 18 men, six women and seven children.
In the three months before their discovery, another 161 illegal immigrants were discovered by Dover Port Police. All had suffered miserable conditions to reach Britain in the hope of a new start. All but seven have since been repatriated.
Port police blame the 'middlemen' for the unnecessary suffering endured by the stowaways. In return for sums of money averaging pounds 2,500 these businessmen offer 'safe' passages which are supposed to by-pass Britain's tight immigration controls and land the stowaway in the allegedly safe hands of a contact who can then put the stowaway in touch with a 'network'.
But the passages are neither safe nor can they be guaranteed successful: stowaways have been crushed to death, frozen to death, starved to death or suffocated.
Dover Port Police have arrested 59 such middlemen in the last three months.
Most have been drivers. The 'king networkers' are more difficult to catch.
They are the ones who lead the stowaways to the containers, unbolt the door for them, help them in, hand them their 'survival kit' (consisting of detailed maps of ports, the names of facilitators willing to help out, food, blankets and knives) then seal the 'tomb'.
It is middlemen too who sometimes 'forget' to tell stowaways to North America that temperatures on the ocean can fall to minus 30C. 'I would estimate that more than 100 stowaways freeze to death each year on the trip to the States,' said Flemming Ramsby, deputy secretary general for the Baltic International Maritime Council Organisation (BIMCO).
This year, four Romanians died at the hands of one such middleman. They died of asphyxiation after they were sealed into a container which had just been fumigated. A fifth survived by gashing out a hole in the container with a knife then holding his mouth against it for fresh air. When Felixstowe Port Police released him, he fell half- dead into their arms. Behind him was the nightmare of every stowaway: a pile of sleeping bags, trainers and faces - all dead, eyes open.
'The middlemen are the real criminals,' said Detective Inspector Tony Masters from Kent Port Police. 'The stowaways are, for the most part, the victims.'
As yet, the Home Office is denying that the high incidence of illegal immigrants smuggling themselves on board ship is a problem. 'We are very satisfied with our port procedures,' said a spokeswoman from immigration, before adding that the Home Office was 'unable to confirm the figures' put forward by Amnesty International (3,400 to 4,500 stowaways this year) or by Mr Ramsby from BIMCO (4,000).
'If someone is found to have entered Britain by stowing themselves on board a ship, that fact is usually held against them during political asylum hearings,' was all she would say. 'An applicant stands a much better chance of being granted Exceptional Leave to Remain if they travel by conventional means.'
The Home Office may be satisfied with its handling of the stowaway problem, but the maritime organisations are not going to let the matter drop. It is the ship owners and crew who bear the brunt of the burden, they say. Not only are stowaways a financial burden (the crew is responsible for feeding and sometimes repatriating them) but they can also be dangerous. 'It is the desperation of these people which scares me most,' said Mr Ramsby.
'Some are so determined to arrive at their destination that they are willing to use force if necessary. Already we have had situations where a stowaway has threatened a crew member with a weapon.'
The IMO says an international agreement is needed. At present ship owners have their hands tied if a stowaway is discovered: current regulations inhibit ship crews from depositing their 'guests' back on land at the first opportunity.
'We need a code of conduct which allocates responsibilities fairly,' said Brian Parkinson who works for the International Chamber of Shipping. 'We hope to have negotiated one within the next three years.'
Sniffer dogs, closed circuit television cameras and stringent identity checks already help to track down stowaways. But the success rate, as police officers are the first to admit, is not very high. 'I would estimate that between 10 to 20 per cent of the illegal immigrants are caught by officials,' said Detective Inspector Masters.
X-ray screening and 'human' detectors are the talk for the future. Ship captains hope to screen containers before they are loaded on to the ships.
But standing on the dock at Felixstowe the task looked daunting; 56,000 containers were in dock that day. Who was to know how many stowaways were already crouched inside this ghost village of orange and yellow steel 'houses'? The containers stacked across the 264-hectare expanse all looked the same.
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