I tinkered around with the monitor a little bit more and discovered
that there's a poor connection on the power supply board. There is a
transformer that makes an audible noise when the board is tapped or
gently moved. With that noise comes a change in the picture -- usually
just in terms of brightness, but I did get a few very bright bursts of
white (presumably due to a slight connection and maybe a spike in
power?).

Other times, though, the monitor powers on like normal, yet I'm still
seeing the horizontal raster lines. The screen control on the flyback
works as it should, and so does the focus.