By Yassin Kombi and Sonia Rolley

GOMA, Democratic Republic of Congo (Reuters) - Dead bodies lay in the streets, gunfire rang out and hospitals were overwhelmed in east Congo's largest city Goma where M23 rebels backed by Rwanda faced pockets of resistance on Tuesday from army and pro-government militias.

A day after the rebels marched into the lakeside city, protesters in the capital attacked a U.N. compound and embassies including those of Rwanda, France and the United States, expressing anger at what they said was foreign interference.

M23 fighters entered Goma on Monday in the worst escalation since 2012 of a three-decade conflict rooted in the long fallout from the Rwandan genocide and the struggle for control of Congo's abundant mineral resources.

Goma is a major hub for people displaced by fighting elsewhere in eastern Congo and aid groups seeking to assist them. Now the fighting has sent thousands of people streaming out of the city - some had recently sought refuge there from M23's offensive since the start of the year.

Democratic Republic of Congo's government and the head of U.N. peacekeeping have said Rwandan troops were present in Goma, backing up their M23 allies. Rwanda has said it is defending itself against Congolese militias.

Goma residents and U.N. sources said dozens of troops had surrendered, but some soldiers and pro-government militiamen were holding out. People in several neighbourhoods reported small arms fire and some loud explosions.

"I have heard the crackle of gunfire from midnight until now ... it is coming from near the airport," an elderly woman in Goma's northern Majengo neighbourhood, close to the airport, told Reuters by phone.

REPORTS OF RAPE AND LOOTING

Jens Laerke, spokesperson for the U.N. humanitarian office (OCHA), told a briefing in Geneva colleagues had reported "heavy small arms fire and mortar fire across the city and the presence of many dead bodies in the streets."

"We have reports of rapes committed by fighters, looting of property ... and humanitarian health facilities being hit," he added. Other international aid officials described hospitals overwhelmed with wounded being treated in hallways.

"The town is a powderkeg," Willy Ngumbi, a bishop in Goma, said. Explosives had hit a house where priests were staying and the maternity ward of a Catholic hospital on Monday, he said by phone. "The youth are armed and the fighting is now taking place in the town."

Francois Moreillon, head of the International Committee of the Red Cross in Congo, told Reuters a medicine warehouse had been looted, and he was concerned about a laboratory where dangerous germs including ebola were kept.

"Should it be hit in any way by shells which could affect the integrity of the structure, this could potentially allow germs to escape, representing a major public health issue well beyond the borders of the DRC," he said.

In Kinshasa, 1,600 km (1,000 miles) away from Goma, angry crowds burned tyres, chanted anti-Rwanda slogans and attacked embassies of several countries seen as favourable to Rwanda, leading the police to fire tear gas.

"What Rwanda is doing is with the complicity of France, the U.S. and Belgium. The Congolese people are fed up. How many times do we have to die?" said protester Joseph Ngoy.

The Rwandan, French, U.S., Ugandan, Kenyan, Dutch and Belgian embassies were targeted. Videos posted online and verified by Reuters showed dozens of people looting the Kenyan embassy, while others showed looting had spread to other locations including a supermarket.

FEAR OF SPIRAL

The U.N. and global powers fear the conflict could spiral into a regional war akin to those of 1996-1997 and 1998-2003 that killed millions, mostly from hunger and disease.

Corneille Nangaa, leader of the Congo River Alliance that includes the M23, has suggested the rebels' aim is to replace President Felix Tshisekedi and his government in the capital .

U.N. peacekeepers have been caught up in the fighting. South Africa said three of its men were killed in crossfire between government troops and rebels and a fourth had succumbed to wounds from earlier fighting, bringing the number of its fatalities in the past week to 13.

M23 is the latest in a string of ethnic Tutsi-led, Rwandan-backed insurgencies that have brought tumult to Congo since the aftermath of the genocide in Rwanda 30 years ago, when Hutu extremists killed Tutsis and moderate Hutus, and then were toppled by the Tutsi-led forces that still dominate Rwanda.

An estimated 1 million Hutus, some of them refugees and others genocide perpetrators, poured into Congo, and Rwanda says Hutu-led militias and their government allies pose a threat to Congolese Tutsis as well as Rwanda itself.

Congo rejects Rwanda's complaints, saying Kigali has used its proxy militias to control and loot lucrative minerals such as coltan, which is used in smartphones.

(Reporting by Yassin Kombi in Goma, Sonia Rolley and John Irish in Paris, Ange Adihe Kasongo and Stanis Bujakera in Kinshasa, Bhargav Acharya in Johannesburg and Giulia Paravicini in Nairobi, Emma Farge in Geneva and Milan Pavicic in Gdansk; Writing by David Lewis and Aaron Ross; Editing by Michael Perry, Andrew Cawthorne, Peter Graff)