If all you have a coin there is a simple algorithm that's equivalent to sorting by random real numbers in [0, 1].

    1. All players flips a coin.
    2. Players that got heads go before players that got tails, forming (up to) two groups.
    3. If a group has more than one player go back to #1 to determine the order within that group.

It's not a finite process though - it could go on forever if really unlucky. But this is unavoidable, since the number of permutations on n players with n > 2 has factors not divisible by 2 there is no finite series of n coin tosses that could without any bias create a permutation, as the number of outcomes is 2^n.