How do I force the 'showing' of an onscreen keyboard (soft-keyboard) when already 'paired' with a bluetooth keyboard? My question is the exact inverse of this question.

Motivating use-case. I've paired and am using my Bluetooth apple keyboard. Just happy. I'm running ICS, on an HTC- Evo 4G LTE. Then, 7-8 mins later (timeout not yet engaged by default of course), I'm downstairs, and a call comes in. I need to type 2-3 letters to look up something (an email address, etc). and huh?!

How the heck do I in the moment on-the-call type?

At that moment don't remember where I exactly left that apple keyboard (which I could  go back to power off, and only then get a keyboard to show)  Or if I turn off Bluetooth entirely: 'Fail' as that often means I can't then continue to use the darn headset I've got on, or if driving, I'd be illegal (in CA need to have a headset).  

If I unpair, seems to get that keyboard (the external Bluetooth apple one) working, I have to re-pair (which, in reality, is really futzy annoying, compared to, e.g. if ignored and the time-out engages, when I then, say a day, or two later, type on that apple keyboard it works instantly instead? not paired).

Regarding the app: "null keyboard" I tried it, and close, but no cigar, meaning: when I'm in a 'text field', in the notifications list, there will appear an 'input method notification' clicking on that and you get a list, and while that list should include something that offers a 'show on-screen keyboard' override, it does not.

The closest thing I've been able to find is developer-centric code examples showing an (apparent) workaround to force this.. but seems to be no app ( yet?!?) that exposes that functionality into the 'inputs methods' drop-down menu. Example.  

Update: 
 it looks like Tomas's answer is 'the answer', though I'm currently unable to test to be sure. This does strongly suggest that there is no answer available without a rooted phone.
You can use Secure Settings Hardware Keyboard Action to toggle the hardware keyboard. Interestingly, you can still use the hardware keyboard, but the on-screen keyboard still appears.

This is especially useful when you connect a mouse (and no keyboard) and it causes the keyboard to go into hardware keyboard mode.

The simplest way to do this is to create a shortcut on your homescreen. Select the Secure Settings shortcut/widget. Then it will prompt you to select the action. Select Actions > Hardware Keyboard. Then it brings you to a screen with three options: On, Off, Toggle.

On means you go into hardware keyboard mode and no on-screen keyboard will appear. Note, this only works if you actually have physical keyboard connected. It will simply give you an error toast if you don't.

Off means turn off hardware keyboard mode, and an on-screen keyboard will appear when you select text boxes.

Toggle means it will switch from whichever hardware keyboard mode it's currently on.

There are other ways to hook up to Secure Settings Hardware Keyboard Action, which you can learn about in the documentation, and using your imagination and research.