To answer my own post here:
The solution was to load the snd-virmidi module that creates virtual raw midi interfaces that can be reached by both Java MIDI (Processing) and the ALSA sequencer MIDI implementation. The default module configuration creates a whole lot of virtual MIDI devices. Create a file /etc/modprobe.d/snd-virmidi.conf containing the following single line of text to configure this:
options snd-virmidi midi_devs=2
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