I don't think the Intel 186 was in any "home computers" or PCs, but I see other people referencing it as a common chip. Also, I've heard that NEC V20s had the 186 instruction set in addition to 8088, etc. That's about the only reference I have to it actually being used. What happened to it and why did the IBM PC and all its clones jump to the 286 chip for a CPU from the 8086?
There was an actual PC (somewhat compatible) computer built with the 80186.
That was the Tandy 2000 from Radio Shack.
I have vague memories of being envious of the better graphics it had over the Tandy 1000 my family owned at the time.
From all that I have read, it wasn't fully compatible with the IBM PC. It did run MSDOS, and many of the commonly used programs were available for it.
The Wikipedia Tandy 2000 page refers to a second computer that I'd never heard of that also used the 80186.
That's the Mindset. It had better graphics than the Tandy 2000, and a MOUSE. It was apparently, though, even less PC compatible than the Tandy 2000.
The 80186/80188 is not a plain microprocessor but a microcontroller as it integrates some peripherals like timers, interrupt controllers and DMA controllers into it.
Compared to an IBM PC which had separate chips for timer, interrupt controller and DMA controller, these 80186/80188 integrated peripherals are either different or at least mapped to different addresses than on IBM PC so you can't build an IBM PC compatible computer with 80186 or 80188.
But a 80286 is again a plain microprocessor so same external timer/interrupt/DMA chips can be connected to it at same addresses than on an IBM PC and the 80286 also had the extended instruction set from 80186/80188. PC compilers usually called these simply as 286 instructions as that was the CPU that introduced them to PCs, in the IBM PC/AT.
"Common" and "common in PCs" are two different things. The 80186 and the 80188 were great chips for embedded application, with -- for the day -- a very high level of integration. For many applications it was the core of the lowest-chip-count solution you could have at the time.
As mentioned, it was incompatible with the PC -- but I'm not sure that's what Intel was thinking when they made it.
It'd be nice to have sales numbers for the 80186/80188 vs. the 8086/8088, just for comparison -- and how many 808x ended up in embedded applications vs. PCs.
Noting that other people have made comments about machines from Tandy and Research Machines, but there was also a very early laptop called the Tava Flier which used it.
Apart from that I agree with everybody else: it was far more commonly encountered in embedded systems and development labs, and was more often than not programmed using Intel's own development systems.
Convergent Technologies put out a business desktop computer, the NGEN, that was initially based on the 80186. This was a follow-on from their 8086-based IWS and 8088-based AWS, and ran an updated version of their CTOS operating system that maintained compatibility with the earlier products. These computers were also marketed by Burroughs as the B25 (following the B21 [IWS] and B22 [AWS], and by several other companies under their own respective brands.
Another 80186-based PC was the Goupil G4, released in 1986 and notable primarily for being one of the very few PCs to ship with the originally-planned multitasking MS-DOS 4 (although the OEM version of the OS that shipped with the G4 was not actually capable of multitasking) - in fact, the G4 was apparently the main reason why Microsoft completed multitasking MS-DOS at all.
CompuPro had add-on CPU cards using the 80186 - SP186
A typical system would have one 16-bit main CPU handling all disk access, running the main operating system (e.g., an 80286 running MP/M-86) and a combination of 8-bit (SPUZ - Z-80) and/or 16-bit (SP186) processors. Terminals could be connected to serial ports on the SP boards, providing a better user experience than a single-CPU system (more total CPU power) and better than a typical network (effectively a network of computers in one box).