I need a glue that won't turn opaque or run
Use a clear liquid epoxy resin, construction adhesive, 'Gorilla Glue', or 'Iron Grip' silicone adhesive:



  Liquid epoxy resins dry clear and hard, bonding stones that may not
  fit snugly together. It may make some stones look wet in the area of
  application. Since this resin is a liquid, it runs a bit after you
  apply it, so it can pool beneath the stones if you use too much. Epoxy
  resin works well on stones, both large and small.



source: smaller scale than what the OP might have in mind, but still a valid example of an end result. Be sure to work safer than the author, though.

As per this site, be sure to warm up the resin before application (though not with a heat gun or similar device), as this may help prevent it from turning cloudy.
In case of outdoor use, be sure to pick an epoxy that is UV stabilized.
Additionally, make sure you apply it in a dry environment and protect it from getting wet (also while curing), and don't overmix it. *  

Update: you won't be able to get blobs with epoxy resin, unfortunately.  


The following glues are commonly or regularly used for glueing brick walls together, or glueing stones to a vertical surface.



  Construction adhesive, offered in a caulk-style tube, creates bonds
  strong enough to hold larger stones together, such as baseball-style
  stones or river rocks around the perimeter of a homemade garden pond.  


Make sure you're using a clear variant.  

source
'Clear Gorilla Glue' (no affiliation): 


  The strength and versatility you have come to expect from Gorilla Glue
  is now available in an easy to use, crystal clear, non-foaming
  formula. Clear Gorilla Glue is incredibly strong and is ideal for
  indoor and outdoor applications and bonds virtually everything.  


See here for an example of mortar replacement. Mind that the regular unclear version is used.
'Iron Grip Adhesive Silicone' (no affiliation): 


  Made from 100% silicone, GE Iron Grip adhesive works fast on almost
  any porous and non-porous surface, including aluminum, steel, PVC,
  glass, concrete, fiberglass, polystyrene, plywood, drywall, ceramic,
  porcelain, stone, laminate, wood and MDF – indoors and out – all
  without any extra surface prep.


Clear curing. Here's a commercial video, showing the glue used on concrete blocks. As always, check the reviews.  



* More useful information on the application of epoxies can be found here.