How do I force my MHL to DVI adapter to send 1080p video to my HD computer monitor?
That idea of adding another converter and going digital-to-analog seems to be one of the ways of getting this kind of setup to work. I added a simple passive DVI-to-HDMI adapter after the above MHL converter (e.g. http://naplestech.com/images/dvi-to-hdmi-adapter.jpg) and then continued with an active HDMI-to-VGA converter, namely this: http://www.sandberg.it/product/HDMI-to-VGA-Converter

Plugged the whole converter-adapter-converter chain into my phone at one end and into the VGA input of the monitor at the other and presto: 720p@60Hz right away.

Now both active adapters have extra USB inputs allowing them to be powered from some source other than the phone, but surprisingly the whole chain can also be powered by the phone alone and will still work. (Everything I've read online about MHL v1.0 says the MHL adapter itself will not work unless powered from an external source, but somehow it seems that's either not true or was not referring to the adapter and the video output but rather to supplying power to the phone while it works in this way.)

Caveats:


Sometimes while streaming the monitor image goes blank for a second or two and then recovers; this seems to resolve as the adapters heat up or somehow "settle in"
Sometimes at the beginning of a usage session the monitor image turns into white noise whenever starting a video or launching a game and does not recover until some part of the chain is unplugged and plugged back in. Not sure what causes this but again it seems to resolve in time, through some sort of "settling in".