Another month brings another Seal o' Piracy for you to pad your trophy page with. And July's limited edition trophy is very simple to obtain--if, perhaps, a little time-consuming. It's back to basics as you obtain your Seal by...
Complete 30 sea battles. Sea battles involving a cutter (either as your ship or the opponent) count triple!
This achievement requires you to return to the activity Puzzle Pirates does best: pillaging! Patrol the high seas in search of ne'er-do-wells and scallywags, fill their hulls full of cannonball holes, and teach their crews a lesson with the edge of your blade or the business end of your bludgeon. Your ships must grapple and you crews must melee for the battle to count toward your Seal.
But if the thought of 30 sea battles sounds like a tall order, pay special attention to the shortcut: Pillage on or against a cutter, and your count goes up by three.
If you want to take a cutter out and go a-hunting, this old guide to soloing a cutter remains useful despite its age. You'll have to adjust for the changes made since then, but most of those changes make the job of pillaging in a cutter much easier. Specifically, the number of swabbies you can have aboard has gone up to 6, and they can be assigned to fill the cannons. (Also, the hold capacity has increased, allowing you to bring home even more booty.)
But of course you're not going to just solo, are you? Advertise your voyage on the Notice Board and spread those Seal-earning pillage-in-a-cutter points around!
What are the chances of your ship encountering a cutter? There are several schools of thought on that.
First, consider the difficulty of your route. The normal way of things is that a sloop will encounter other sloops. But if you're on a particularly "red" route, like the Ruby or Osprey archipelagos, you stand a chance of getting into it with a cutter. (Or with a dhow or a fanchuan.) Again, these are old threads, but your Examiner's experience bears out their continuing validity. Meanwhile, you can consult the consensus of the Yppedia editors on the difficulties of all routes on the Cerulean, Emerald and Meridian Oceans.
Next, consider the ship you're in and how fully you've staffed them. MaxPain and BobJanova have compiled their own lists of what ships you're likely to encounter depending on the ship you're sailing. They both agree that dhows, fanchuans and cutters constitute a discrete class of spawn type, such that they will naturally spawn each other. Max your sloop capacity and you may see the cutter-inclusive type of spawn as well.
Lastly, don't overthink it. Just get out there and pillage for fun and profit, and the Seal will take care of itself.