Glue for polystyrene
Polystyrene is attacked by many organic solvents, so I'd rule out all solvent-based glues you haven't personally tested.  Different compositions can be sold under the same name in different countries, or can change over time, and very similar names can have different ingredients.

That leaves you with two options; I recommend the second: 

Water-based glues: These will give you a low-strength bond on polystyrene, but if you have a large area of a fairly  weak material that might not matter.  PVA (white craft glue, wood glue) would be one type, latex (e.g. Copydex brand) would be another, but you'd need to be sure you were getting a water-based  product (many other glues with latex/rubber in the name are solvent-based).  You'd need to use a thin layer, and drying times would be longer than expected, because you're relying on water  evaporating and it has to escape.

Hot glue: Cool melt hot glue is plastic, which is melted and applied to one surface.  This sticks well to polystyrene.  It cools and solidifies fairly quickly when exposed to air, but extruded polystyrene will keep the heat in, so setting times will be increased.  As with water-based glues you'll have to support the workpiece while the glue dries.  This should give a stronger bond, as it is better able to fill the gaps in the surfaces.  You can get a range of colours, but don't expect to colour-match.  With decent pressure as the joint sets you can avoid gluing right to the edge; with a suitable colour the join can be subtle.