Novel reconfigurable System-on-Chip (SoC) devices offer combining software with application-specific hardware accelerators to speed up applications. However, by mixing user software and user hardware, principal programming abstractions and system-software commodities are usually lost, since hardware accelerators (1) do not have execution context —it is typically the programmer who is supposed to provide it, for each accelerator, (2) do not have virtual memory abstraction —it is again programmer who shall communicate data from user software space to user hardware, even if it is usually burdensome (or sometimes impossible!), (3) cannot invoke system services (e.g., to allocate memory, open files, communicate), and (4) are not easily portable —they depend mostly on system-level interfacing, although they logically belong to the application level. We introduce a unified Operating System (OS) process for codesigned reconfigurable applications that provides (1) unified memory abstraction for software and hardware application parts, (2) execution transfers from software to hardware and vice versa, thus enabling hardware accelerators to use systems services and callback other software and hardware functions, and (3) multithreaded execution of multiple software and hardware threads. The unified OS process ensures portability of codesigned applications, by providing standardised means of interfacing. Having just-another abstraction layer usually affects performance: we show that the runtime optimisations in the system layer supporting the unified OS process can minimise the performance loss and even outperform typical approaches. The unified OS process also fosters unrestricted automated synthesis of software to hardware, thus allowing unlimited migration of application components. We demonstrate the advantages of the unified OS process in practice, for Linux systems running on Xilinx Virtex-II Pro and Altera Excalibur reconfigurable devices.