Well, you can use without doubt an FPGA without software, in fact many are used to support booting more complex systems (power sequencing, for example). But you can also define some kind of embedded soft processor needing an OS.
On the other hand no CPU really requires an operating system, you could probably program a Cortex-A (for example) bare to the metal. The operating system/user software is only a useful distinction in responsabilities.
There are also ASIC which are single task but are however microcoded with some mask defined program to do their job (some USB bridge comes to mind).
In short anything can go
As for the real time issue: there's nothing more real time than a state machine on the FPGA fabric (except maybe a real ASIC), every kind software is slower (since it's by definition interpreted and run on a state machine in the FPGA fabric)