I don't know how much you understand about these two procedures, and I apologize if what I write below is redundant.
The only way you're going to get a good answer to your question, is to define the word, "tighten." Tighten the fastening of the fundus against the esophageal wall? (Perhaps to make the Nissen more secure against slippage?) Tighten the action of the Nissen? (Presumably because it's not stopping enough reflux?) I tried to address both concerns in my post above, as its use in this respect is somewhat ambiguous, and impossible to address specifically.
If your surgeon has never had to fix his own procedures, why is this on the table?
Also, and I'm presuming this is for your own case, have you had a post-op pH test? (This is also presuming that the Nissen is not stopping the reflux that it should, and therefore needs "tightening.") The theoretical limit of a Nissen is 2.0% pH<4.0. The lap Nissen does not stop all reflux, even when it's performing "perfectly." Any attempt to make it perform better than this 2.0% number is misguided.
I understand that it's difficult to visualize what's going on with a lap Nissen or a TIF from a textual description. If you can't approximate it using both hands in front of you to help visualize, try finding images or video. Because of the way the Nissen is wrapped around the esophagus, it makes most of the circumference of the wrap inaccessible to the EsophyX device, especially if natural post-surgical attachment has taken place. It's possible that the doctor could've used the term, TIF, in an off-handed way, meaning that his intention was to use the EsophyX device (or even another type of surgical stapler) in an UNCONVENTIONAL manner to make attachments. It would not be correct to call this a TIF procedure, but simply a means to use the EsophyX device as a stapler.
As I pointed out in my post above, the "tightness" of the lap Nissen is determined by the wrap. No amount of stapling through the esophageal wall, even if it were possible 360 degrees, would improve upon this. Even if it were possible, all it would accomplish is securing the fundus more tightly against the esophagus...it would do nothing to improve the peristaltic waves of the fundus, which is what presses the LES closed.
It's these same peristaltic waves that cause reflux in the first place, when the LES is weakened. Therefore, using the fundus around the LES in a lap Nissen, it replaces the LES function somewhat, and also insures that the initial peristaltic waves clamp the LES shut first before the waves move through the stomach.
The stand-alone TIF is inferior to the lap Nissen, because the TIF only lifts and secures the fundus against the esophagus, and the surgical site only covers perhaps one centimeter of free-hanging esophageal tissue into the stomach, plus another 1.5 cm. of stapling above it. (Also, because of the limitations of the device to be able to rotate, it can't complete a full 360-degree procedure.) The Nissen perhaps doubles the size of this surgical site. The additional benefit of the Nissen is the pressure of the wrap around and against the LES...but again, this pressure is due to the tension of the lap laterally around the esophageal wall and secured with the four sutures. The exact tension is determined by the surgeon through the use of a bougie inserted down the esophagus as he is suturing.
This is pure conjecture on my part, but if the surgeon inserted a surgical stapler through one of the abdominal entry ports to gain access to the "front/right" section of the wrap where the sutures are located, grasped the two ends of the fundus wrapped around the esophagus and pinched them more closely together -- then stapled them fundus to fundus -- this would make a tighter wrap. This has no resemblance whatsoever to a TIF procedure (as the TIF is done transorally), but it could use H-fasteners and a stapling action similar to a TIF. This would not work on a Toupet, because the ends of the fundus are intentionally sutured against the esophageal wall 90 degrees apart.
I'm sorry, but that's about all I can say about your question. You'll have to talk to him to clarify the definition, and his answer. One other comment about the EsophyX device: it is a single-use instrument, and costs nearly $5,000. Fewer and fewer insurance companies are offering the coverage for this equipment purchase.