By Tom Finn

CAIRO (Reuters) - A few lonely Egyptian activists are trying to stake out middle ground in the nationwide rift between an affronted Muslim Brotherhood and delighted supporters of the army's overthrow of President Mohamed Mursi.

Those rival camps have filled the streets with myriad demonstrators since General Abdel Fattah al-Sisi ousted and jailed Mursi, Egypt's first freely elected leader, on July 3.

Alarmed by the polarization and alienated by both sides, some Egyptians started up a "Third Square" movement promoting a middle way to avert conflict in the most populous Arab state.

"We are stuck between two bad options: an army killing without reason and an intolerant Islamist movement which wants a theocratic state," said Tariq Ismaeli, a 34-year-old civil engineer in jeans and red sneakers at a rally on Sunday.

"We are trying to establish a new voice," he said.

The rally in Cairo's Sphinx square drew about 300 liberals, leftists and moderate Islamists dismayed by Saturday's carnage, when security forces killed 80 Muslim Brotherhood partisans in clashes at a protest camp set up to demand Mursi's restitution.

Ismaeli and his colleagues have used Facebook and Twitter to marshal several rowdy demonstrations in downtown Cairo, seeing themselves as heirs of the popular uprising that ended Hosni Mubarak's 30 years of one-man rule in 2011.

Their banners carry the faces of Mursi and Sisi crossed out in red and a blunt message: "Topple all who betrayed us. No to theocracy. No to the military junta. Yes to a civil state."

They hope Sphinx Square will become the kernel of a new opposition movement, though their numbers are dwarfed by the Brotherhood's month-old vigil and one manned round the clock in Tahrir Square by supporters of the army's intervention.

"LUNATICS AND DREAMERS"

Third Square is attracting some attention, if mostly online.

Its Facebook page wins around a thousand new "likes" a day and is bedecked with jokes and cartoons lampooning political leaders - a contrast to the reverence of their rivals.

"They've created a space where the original attitude of the revolution expresses itself, where the aims of the revolution are remembered. They're keeping an ember alive," Ahdaf Soueif, an Egyptian novelist endorsing the campaign, told Reuters.

But for now, their message is being swept out of public view by the sheer numbers of Egypt's main opposing camps.

When those at Third Square's first protest last week heard that a huge pro-Sisi march was approaching the venue en route to Tahrir Square, they scattered into the night fearing a fight.

Tamarod, the group that organized mass anti-Mursi protests and now supports the army-installed interim government, accuses Third Square of splitting Egypt's "revolutionary forces".

"I see in this moment Third Square dividing the people. They are living in the past. Now is the time for consensus, we need to move forward," Tamarod spokesman Mohammed Abdul Aziz said.

Pro-Mursi demonstrators camped near a mosque in northeastern Cairo say they will tolerate other protest movements as long as they oppose the military's plunge into politics.

"Even if Mursi doesn't come back, at least the army has two squares to deal with," said Abdulrahman Daour, a spokesman at the Brotherhood camp. "I don't mind people opposing Mursi. Just don't betray the revolution. Don't give power to the military."

Third Square may have little impact on the Brotherhood and the army, Egypt's two most organized institutions, analysts say.

"I don't think they will get much traction. The country is too polarized. It's a zero sum game," said Adel Abdel Ghafer, an Egypt scholar at the Australian National University.

"You are against the army which has the monopoly on violence... and the Brotherhood which has a religious legitimacy argument. So you're basically against everybody."

However, Third Square organizers point to the power of ideas.

"Back in 2011 when we started protesting against Hosni Mubarak, we were insulted. They called us lunatics and dreamers," said 30-year-old Ahmed Nasr.

"It's not about the numbers, it's about the cause."

(Editing by Noah Browning and Alistair Lyon)