The phrase "effective majority" means having more than half of the people or votes necessary to make a decision or take an action. It refers to a situation where there are enough supporters to achieve a desired result. Full definition
Even if a politically effective majority of citizens were Catholic, we should not want a confessional state if that means the Church, in its hierarchical self «definition, would be party to executing the necessarily coercive power of the state. (firstthings.com)
We might ask: In a society where everybody, or at least a politically effective majority, accepts the authority of the Bible, why shouldn't we be able to appeal explicitly to biblical teaching in political disputes? (firstthings.com)
The elections were expected to be much closer, but Cameron's Conservative Party actually had a stronger showing than they did in 2010, winning an effective majority in the Parliament. (relevantmagazine.com)