BAGHDAD—Iraq's vice president ordered parliament to convene on Tuesday to begin the process of electing a new government, a step the Obama administration considers crucial to bridging the deep fissures among the country's religious and political factions and quelling an extremist insurgency.

Prime Minister Nouri al-Maliki, a Shiite, faces mounting pressure from home and abroad to step down and make way for a government that encompasses Sunni and Kurdish politicians.

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