Some editors have this. It's less obstrusive than I thought it would
be.
* lisp/progmodes/flymake.el (flymake--delete-overlay): New helper.
(flymake--highlight-line): New flymake-eol-face considered.
(flymake--clear-foreign-diags): Use flymake--delete-overlay.
(flymake--publish-diagnostics): Use flymake--delete-overlay.
(flymake-mode): Use flymake--delete-overlay.
(flymake-error-eol)
(flymake-warning-eol, flymake-note-eol): New faces.
(flymake-show-eol-overlays): New option.