After my talk at the Berkeley math/physics meeting, I talked a little with Davide Gaiotto.  This resulted in the following compelling pictures which I do not actually understand:

1. Say you have a spectral network and a spectral curve.  There’s an exact Lagrangian naturally associated to this data: each line of the spectral network comes labelled with two sheets of the cover; make the exact Lagrangian which stretches in cotangent fibers between the lift of this line to those sheets.We can’t use this prescription for two reasons: one, the Lagrangian doesn’t go to infinity; two, it is singular along the junctions of the spectral network.  But maybe it can be canonically smoothed at these junctures, and then canonically pushed to infinity along cotangent fibres.  Note it already comes with a bounding knot.
2. One might ask the reverse question — a spectral network lifts to some picture on a spectral curve; what might this have to do with a bounding knot at infinity?  (Recall that via NAHT, the link of the spectral curve near the punctures is the knot we are considering.)  So, take that knot and “use the data of the spectral curve to flow it inward from infinity”.  At some point it will get caught on itself, maybe this is the spectral network.