BENGHAZI, Libya — A month after the killing of the U.S. ambassador ignited a public outcry for civilian authority over Libya’s fractious militias, that hope has been all but lost in a tangle of grudges, rivalries and egos.

Taming the militias has been the threshold test of Libya’s attempt to build a democracy after four decades of dictatorship under Moammar Gadhafi. But how to bring them to heel while depending on them for security has eluded the weak transitional government, trapping Libya in a state of lawlessness.

The dynamic is making it difficult for either the Libyan authorities or the United States to catch the attackers who killed Ambassador Christopher Stevens.

Shocked by that assault, tens of thousands of people filled the streets last month to demand the dismantling of all the militias.

But the country’s interim president, Mohamed Magariaf, warned them to back off as leaders of the largest brigades threatened to cut off the vital services they provide, like patrolling the borders and putting out fires.

Now that problem has become entangled in the U.S. presidential race as well, with Republicans arguing that the Obama administration’s failure to protect Stevens illustrates the unraveling of its policy in the region.

Mounting pressure on the administration to act against the perpetrators carries its own risks: A U.S. strike on Libyan soil could produce a popular and potentially violent backlash in the only Arab country whose people largely have warm feelings toward Washington.