phrase requests - Word for having strong political opinions but no consistent "side"

(With your call for informal words/phrases as well in mind, you could consider the following):

cf. “Politics makes strange bedfellows” (from Dictionary[dot]com)

If the person you describe ‘awakens’ often enough with the proverbial “strange bedfellows” because of his/her across-the-spectrum opinions, perhaps it wouldn’t be too far-fetched to label them as
“a strange bedfellow” themselves (after all, it takes two to tango, no?)
(example from Latin Times);

or to avoid a possible double entendre:

a/the proverbial strange bedfellow.

To avoid coming right out and calling them a “strange bedfellow,” even a proverbial one, perhaps you could use the term as an adjective (one that I would hyphenate) to describe their political opinions or sympathies, i.e., their politics:

“[His/her strange-bedfellow politics … cut across familiar partisan and ideological lines.”
(example of unhyphenated usage from ‘Debating Immigration’ by Carol M. Swain, via Google Books)