This opened up a fundamental rift between IBM and Microsoft which would doom the OS/2 project. Believing that no multitasking OS could ever be made to work on the 286, Microsoft secretly began working on its own successor-to-DOS project, called "Windows."
While Windows 1.0 was targeted to the old 8088 processor, by 2.0 they were offering a "Windows/386" version that was targeted at Intel's new chip. They offered a "Windows/286" version of Windows 2.0 as well, but it didn't even try to use the chip's protected mode. It just treated it like a slightly faster 8088 with a few modest extra features.
In 1985, Intel finally released its follow-on to the 286, the 80386. And the 386 solved all of the 286's technical problems.
Beyond being able to switch back and forth between protected and real modes without rebooting, the 386 also added a new "virtual 8086 mode," in which software that could only run in real mode could be run inside a little sandbox that, to the code, looked just like an 8086 in real mode. Now all your old DOS programs could run right alongside your shiny new OS/2 or Windows ones.
The 386 was exactly what programmers and users wanted. (Which is why, even today, lots of software is compiled to use the 386 as its baseline platform. Every CPU that Intel released afterwards is, effectively, a 386+.)
In 1985, Intel finally released its follow-on to the 286, the 80386. And the 386 solved all of the 286's technical problems.
Beyond being able to switch back and forth between protected and real modes without rebooting, the 386 also added a new "virtual 8086 mode," in which software that could only run in real mode could be run inside a little sandbox that, to the code, looked just like an 8086 in real mode. Now all your old DOS programs could run right alongside your shiny new OS/2 or Windows ones.