That is a thing, although not to the read-level. I've seen it used in bisulphite mapping where Bismark, PASH, and one other (but i forget the name) where used to map reads and the combined methylation calls were used.
The problem is that it doesn't always statistically make sense. It depends on the power/sensitivity of the tools, which is often unknown, and the size of the data. To help explain this a bit better i wrote you a little tool - although it might have a bug. i dont know. it's a sunday and i seem to write the most bugs on sundays.
gist.github.com/JohnLonginotto/bd1a846544b31ebe6f96190c1cd84a83
And you provide it arguments like:
python ./should_i_intersect_it.py --true_positives 100 --true_negatives 60000 --true_positives_A 34 --false_positives_A 10 --true_positives_B 40 --false_positives_B 13
And then it will tell you True Positive, True Negative, False Positive, and False Negative for each tool, and the intersect/union:
Tool A Tool B Intersect Union
TP TN FP FN TP TN FP FN TP TN FP FN TP TN FP FN
34.0 59990. 10.0 66.0 40.0 59987. 13.0 60.0 13.62 59977. 0.0 39.62 60.38 60000. 23.0 86.38
34.0 59990. 10.0 66.0 40.0 59987. 13.0 60.0 13.16 59977. 0.0 39.16 60.84 60000. 23.0 86.84
34.0 59990. 10.0 66.0 40.0 59987. 13.0 60.0 13.74 59977. 0.0 39.74 60.26 60000. 23.0 86.26
34.0 59990. 10.0 66.0 40.0 59987. 13.0 60.0 13.1 59977. 0.0 39.1 60.9 60000. 23.0 86.9
34.0 59990. 10.0 66.0 40.0 59987. 13.0 60.0 13.82 59977. 0.02 39.82 60.18 59999. 22.98 86.18
The above example values i chose because that sort-of reflects RNA-seq differentially expressed genes. Where you have 100 truly DE genes, 60000 genes that don't actually change, and then two tools, A and B, where A gets 34 right but 10 wrong, and B gets 40 right but 13 wrong.
The result is that neither the intersect nor the union look particularly appealing. It depends on what you're after really.
There are values where an intersect or union look very appealing, but meh. Play with the values to figure it out :)
Also run it in pypy because i didn't optimise anything and it's real slow.
EDIT:
GAH Biostars bot you **er, i just burnt 2-3 hours writing a stupid tool for a 4.5 year old question D':
Good suggestion, I'd be interested in seeing the results as well