A solution to this problem exists because there are an even number of both heads-up coins and tails-up coins, and thus they can be split evenly into two piles. If a solution exists, then all you would need to do is arrange the coins into two piles, and randomly swap one coin from each pile, over and over. Given enough time, you would generate all possible arrangements of coins, and at least one is a solution to the problem. Since the original question doesn't specify that you need to leave the coins in a solved state, you would have found the solution, you just wouldn't know when it happened. But, it can be done.
There are something like 9.332E157 total permutations of arrangements, but I believe there are only 11 discrete states, so it's likely that no human could ever do this, but an un-described player definitely could!