The woman was interviewed by police for five hours in hospital last night, the Herald understands.
Victim Support was also working with the young woman, helped by a translator as she did not speak much English.
News is starting to spread around the local Czech community today.
New Zealand is a popular backpacking destination with the Czech Republic. It gets allocated 1200 12-month travel visas every year. More than 30,000 applied for the limited places this year.
One of the successful applicants was 33-year old Martin Fucik from Ostrava.
The main attraction is New Zealand's spectacular wilderness, he told a Herald reporter in the nearby Queenstown.
"The big attraction is the outdoors. It's very famous now in my country," he said.
"We have small mountains and many Czechs like looking for adventure."
News of the tragedy would be reported widely in his homeland, Fucik said.
Many Czechs work in Queenstown as chefs.
But nobody spoken to today so far knew of the couple.
Some locals expressed surprise that the track was attempted in such wintry conditions.
But one local hunter believed that, despite recent snow dumps, it would've possible to walk out and raise the alarm.
"It's all very strange," the hunter said.
Heliworks general manager Richard Mills said the woman was "ecstatic" to be found.
The Department of Conservation was supporting the young woman, working with police to figure out exactly what had happened.
"This is a tragic incident and our condolences and sympathies are with the families of this couple," DOC Wakatipu operations manager Geoff Owen said.
Until the police have spoken to the young woman, DOC could not comment further on the specifics, Mr Owen said.
"There are significant hazards walking our southern Great Walks during the off season because of the winter weather conditions, avalanche risk and reduced facilities."
Huts on the track are open but wardens' quarters, where the woman was found, were locked.
Read more:
• Routeburn Track 'hazardous' in winter
Huts were not regularly checked during the off season, Owen said.
Concerns were raised with police yesterday morning that a couple who planned to walk in the Routeburn Track area had not been heard from since late July.
The couple's car was located at the track's carpark and appeared to have been untouched for some time.
An operation to find the missing tourists was mounted yesterday afternoon with Search and Rescue and police flying into the area by helicopter.
Otago Lakes Central Area Commander Inspector Olaf Jensen said the woman was taken to hospital for assessment and was in good health although upset by these events.
The Lake Mackenzie hut is about 43km away from the Glenorchy end of the Routeburn track and takes about 5 hours to reach from the Divide on Milford Rd.
Jensen described the circumstances surrounding the death and that nearly a month had lapsed before the alarm was raised as "highly unusual".
"It very unusual for someone to be missing in the New Zealand bush for such a long period without it being reported.
"I appreciate there are a number of unanswered questions, however, until we can piece together exactly what has happened we are unable to say anything further."
Police expected to speak to the woman later today and would be working with her to establish what happened.
This morning Otago Tramping and Mountaineering Club spokesman Ian Sime said it was unlikely the couple had left tramping intentions or signed in before starting their hike.
He said it normally just three days to complete the popular walking track.
"She can't have filed an intention form at the start because if she did someone would have been looking for her within a week."
"Five weeks - it's just unbelievable," said Sime.
Given she had stayed in a warden's hut she would have been warm and had access a large supply of food.
"If she was in the warden's hut she would have been okay," he said.
The Lake Mackenzie warden's hut was locked at this time of year so the woman must have broken in, Sime said.
There were no conservation rangers stationed along the track outside the Great Walks season.
The Great Walks season ended on the 27th of April this year and would restart on October 25 according to DoC's website.
Sime said at this time of year it was possible snow conditions would make the track difficult for people to pass through.
Flushing toilets at the site were shut down for the off season, replaced with pit toilets and gas was not provided, with DoC recommending online that trampers bring their own cooking stoves.
Running water was also turned off inside the huts but could be accessed from the outside water tank or by melting snow.
Ultimate Hikes general manager Noel Saxon, who runs guided tramps of the track, was surprised it took a month before the woman was found.
He said even during the quieter winter season he would have expected another tramper to have come across her earlier.
Heliworks Queenstown Helicopters general manager Richard Mills said his crew flew the woman out of the area.
He said the woman was "ecstatic" when she was rescued.
There had been no instructions about recovering the man's body.
He said the tragedy could have been averted if the couple were carrying a personal locator beacon.
"If any good can come of this it is the message to carry PLBs. They can take the 'S' our of SAR [Search and Rescue]."
The Routeburn Track stretches for 32 kilometres linking the Mount Aspiring and Fiordland National Parks.
It can be entered at either end with one entrance at Glenorchy and the other at the Te Anau-Milford Rd.
It is ranked as an intermediate track by the Department of Conservation.
Police and SAR would be returning to the track this afternoon to try and find her dead companion depending on conditions.
MetService meterologist Claire Flynn said the area was hit with heavy snow from the day the pair entered the track through to the 28th, when the man slipped and died.
Flynn said the average temperature between July 24 and today has been 5.5C at Milford Sound Airport. As the airport is at sea level, it will be colder at higher altitudes.
An experienced tramper say the story of a woman sheltering in a hut for a month after her partner fell to his death on the Routeburn Track is bizarre and hard to believe.
Ultimate Hikes general manager Noel Saxon said at this time of the year there's likely to be a lot of snow, markers would be covered and walking out would be very difficult.
Mr Saxon said it's remarkable the woman survived a month, and that it would've been a very lonely experience.
Questions the Herald has asked DoC they have yet to respond to:
• How was the alarm raised?
• Where did the search and rescue team enter the area from?
• How could she have stayed in the hut for five weeks without anyone coming across her? Or did people come across her?
• Were there any intentions left with DoC by the couple?
• What have the conditions been like in the area around the hut she's been in?
• Has the Routeburn been impassable in the time she's been there?
• What was in the warden's hut in terms of food supplies, blankets etc?
