* Interim PM rows back from proposal to ban Brotherhood
* But Beblawi says Islamists must "face reality"
* Speculation grows of army olive branch to Brotherhood
* Mursi's supporters to try to relaunch protests (Adds interim PM's comments to Egyptian newspaper)
By Lin Noueihed
CAIRO, Aug 28 (Reuters) - Egypt should not ban the Muslim Brotherhood or exclude it from politics after the army's overthrow of Islamist President Mohamed Mursi, the interim prime minister said, reversing his previous stated view.
The apparent about-turn fuelled speculation that the military-installed government may now seek a political settlement to the crisis, but also coincided with a new call for protests by Mursi's supporters.
Hazem el-Beblawi, the interim prime minister, had proposed on Aug. 17 that the Brotherhood, the Arab world's oldest and arguably most influential Islamist group, should be dissolved, and said the government was studying the idea.
In an interview with state media late on Tuesday, Beblawi appeared to row back, saying the government would instead monitor the group and its political wing and that the actions of its members would determine its fate.
"Dissolving the party or the group is not the solution and it is wrong to make decisions in turbulent situations," the state news agency MENA quoted Beblawi as saying.
"It is better for us to monitor parties and groups in the framework of political action without dissolving them or having them act in secret."
But he tempered his comments in a separate interview with the newspaper al-Shorouk, saying parts of Egyptian society "think that the Brotherhood does not truly desire reconciliation", and urging it to "face up to reality".
The government has portrayed its attack on the Brotherhood as a fight against terrorism, and Beblawi said ordinary citizens were "afraid of reconciliation with people who use force".
There has been no sign from the Brotherhood, most of whose leaders are now in jail or on the run, that it wants to engage with the army establishment that bulldozed it out.
OUTLAWED FOR DECADES
Founded in 1928, the Brotherhood was banned by Egypt's then military rulers in 1954. Though still outlawed during the 30-year rule of Hosni Mubarak, it ran a large welfare network and its members ran as independents in limited elections.
After decades of operating in the shadows and winning support with its charities and preaching, the Brotherhood registered itself as a non-governmental organisation in March in response to a court challenge by people contesting its legality.
It also has a registered political arm, the Freedom and Justice Party (FJP), set up in 2011 after Mubarak's overthrow in an uprising. The Brotherhood won all five national votes held since 2011, including Mursi's election as president last year.
But Mursi alienated swathes of Egyptians during his year in power and, after mass protests, the army removed him on July 3.
More than 1,000 people, including about 100 police and soldiers, have since been killed in the worst internal violence in the Egyptian republic's history. Most died when the security forces dispersed two pro-Mursi protest camps on Aug. 14. State media have described the crackdown as a war on "terrorism".
With the Brotherhood in shock, protests that it called last Friday mostly failed to materialise.
The National Coalition to Support Legitimacy and Reject the Coup, which includes the Brotherhood and demands Mursi's reinstatement, promised protests in the streets and squares of all of Egypt's 27 provinces this Friday and said it would "activate a plan of peaceful civil disobedience".
The newspaper run by the Brotherhood's FJP ran an advertisement under the banner "Boycott the murderers", urging citizens not to watch the television channels of the "old regime"; not to pay taxes to "the government that is killing us"; and not to buy products made by companies that it accused of supporting the military-backed interim government.
The government says it will call parliamentary and presidential elections within months, after the passage of a new constitution. (Additional reporting by Maggie Fick; Editing by Kevin Liffey and Alison Williams)