By Jenny Clover

KIGALI (Reuters) - The U.S. embassy in Rwanda was working out on Tuesday what to do with a Congolese warlord wanted by the International Criminal Court, a day after Bosco Ntaganda walked off the street and turned himself in to face war crimes charges.

Ntaganda stunned U.S. embassy staff when he walked into the diplomatic mission and gave himself up, an apparently meek end to a 15-year long career that saw him fight as a rebel and government soldier on both sides of the Rwanda-Congo border.

He specifically asked to be transferred to the Hague-based tribunal, the U.S. State Department said.

Neither the United States nor Rwanda has an obligation to hand over the commander nicknamed "the Terminator" to The Hague-based ICC as they are not signatories to the Rome Statute that set up the court.

Washington broadly supports the ICC, but testimony by Ntaganda, who has fought in a string of Rwanda-backed rebellions in Congo's east, may be damaging for the government of Rwandan President Paul Kagame, a close U.S. ally.

"I don't think we have any real idea of when things will happen," an official at the U.S. embassy in Kigali told Reuters. "We are still figuring out how it's going to work."

Rwandan-born Ntaganda faces charges of recruiting child soldiers, murder, ethnic persecution, sexual slavery and rape during the 2002-2 conflict in northeastern Congo's gold-mined Ituri district.

But a U.N. panel of experts said Ntaganda was most recently a leader of the year-old M23 rebellion - an insurgency in eastern Congo the experts said was backed by senior Rwandan government and military officials.

"He's on the compound and we have a place for him to sleep but obviously we're not a hotel and we don't have a guest room," the embassy official told Reuters.

"NOT RWANDA'S BUSINESS"

One of Africa's most wanted men, Ntaganda's defeat came after a split within the M23 rebel movement in past months left the rebel commander increasingly sidelined.

A rival, more powerful faction of M23 had shown signs of warming up to a peace deal with the Kinshasa government.

With an international arrest warrant hanging over him, Ntaganda worried he might be sold out as part of any peace deal and was seen as a potential spoiler to the process, Jason Stearns of the Rift Valley Institute wrote in a briefing paper days before Ntaganda's surrender.

Rwanda has said the decision whether to transfer Ntaganda to the Hague is not its to make.

"Congolese citizen, U.S. territory, going to the Hague is not Rwanda's business," Rwanda's Foreign Minister Louise Mushikiwabo said on Twitter late on Monday

Ntaganda's removal from the battle theatre raises hopes of improved stability in a region where colonial era boundaries cut through ethnic groups and local politics and vast mineral resources have fuelled two decades of conflict.

Rwanda and Uganda, which the U.N. experts also accused of supporting the M23 rebellion, have both repeatedly denied involvement. Both countries signed a U.N.-mediated deal signed last month aimed restoring peace and stability to Africa's Great Lakes region.

Wars in the central African nation have killed about 5 million people in the past decade and a half and many eastern areas are still afflicted by violence from a number of rebel groups, despite the decade-long peacekeeping mission.

Children have been conscripted into rebel ranks and rape has been used as a weapon of war, rights groups say.

"We repeat that people responsible for these war crimes should be brought to justice," U.N. human rights spokeswoman Cecile Pouilly told a news briefing in Geneva on Tuesday.

(Additional reporting by Stephanie Nebehay in Geneva; Writing by Richard Lough; Editing by Edmund Blair and Alison Williams)