When using a turnbuckle, there is no need to develop a large pulley system to provide the mechanical advantage to provide tension thanks to the wonderful turnbuckle located in line! Turnbuckles are compact wonder machines, containing a very long inclined plane as well as lots of leverage to push the load up that plane. Inserting a long screwdriver into the turnbuckle body will add a lot of leverage to the system and allow you to achieve the necessary force to tension the rigging.
The reason turnbuckles can't be used alone with synthetic stays is they don't have enough travel to absorb the stretch that will occur over the life of the stay. Turnbuckles offer five inches of travel, and once that is consumed, you can no longer tension the stay further. Lashings on the other hand offer a lot of travel and can absorb the stretch and creep of a synthetic stay over its life.
To tension synthetic standing rigging with a turnbuckle, all you need to do is connect the turnbuckle to the deadeye, the deadeye to the lashing, and the lashing to the stay. With the turnbuckle as loose as it can be (just one thread showing on the top and bottom screws), tighten the lashing using a marlin spike hitch as much as you can by hand, then tie a shroud frapping knot in the lashings to lock them in place.
With the lashings secure, begin tightening the turnbuckle until the desired tension is achieved. If and when the turnbuckle becomes two blocked, which will happen, simply loosen the turnbuckle completely and re-tighten lashings again using a marlin spike hitch. Simply tie a new shroud frapping knot and re-tighten the turnbuckle.
You might think that the stay will be just as tight as it was when you started, but in fact it will be very slack. In essence, the stay that was pretty tight while two blocked will now be five inches longer when the turnbuckle is relaxed. These five inches will then be absorbed by the lashings and tension can be re-introduced via the turnbuckle once again.
It might seem like a convoluted procedure, but trust me, it is much simpler than running tensioning lines on the deck to a cockpit winch and setting fair leads at the perfect position to tension your deadeyes. Turnbuckles offer excellent ease of adjustability in a compact package for an equally significant price.