Many adhesives will probably do the job, such as tough acrylics and epoxies. The simplest to source and start with might be a clear 5-min epoxy, dispensed from a twin-syringe and fitted with a static mixer. As its name implies, this type of adhesive will give good handling strength in 5 min and will cure completely in less than 24 hrs at room temperature. If desired, the curing can be accelerated with mild heating in an oven or by using a heat lamp. Its adhesive strength may exceed the strength of the polystyrene, and this type of epoxy does have some toughness to absorb impacts.
If you need something that cures even faster, versions are available that fixture in 30 sec. With these products, you will need to keep dispensing the adhesive quickly to avoid it setting up in the static mixer. This type of system is extremely easy to use and can be automated in the future if your production requirements increase considerably.