You would do permutation testing here. First randomly assign each trial to condition A or B, and then get the average for condition A - condition B. Repeat this process a thousand or so times to create your null hypothesis distribution. This should have a mean of around 0.
Next evaluate the significance by
zValue = (yourObservedDiff-mean(nullDiffs))/std(nullDiffs)
pValue = (# of nullDiffs > yourObservedDiff)/(total # of nullDiffs)