Hi Nathali,
In practice, it's quite difficult to compare samples in the standard Qiime pipeline that have been generated with different primer sets, even when the sequences overlap. Using a typical de-novo clustering approach will almost always yield separation by primer pair rather than by meaningful biological categories, especially if (as in this case) one primer set is specific to a particular group of bacteria.
What are your specific questions? Is there another approach that might be useful?
Just to throw out some ideas:
1) closed-reference OTU picking against a common database (like GreenGenes) could give you some idea of the relative diversity covered by the two sets, even in the absence of any trimming.
2) clustering using you general bacterial data and then closed-reference picking with your cyanobacterial data against those clusters could give you a specific idea of which cyanobacteria are being picked up by your general primers.
3) the script '
truncate_reverse_primer.py' could be used in conjunction with one of the universal primers to remove the overhanging tail of the cyanobacteria-specific sequences, and then these sequenced used in open-reference OTU picking against your universal bacterial sequences.
Cheers,
-jon