From MSW forum member Wefalck (France): A bullseye is meant to serve as guide to redirect a run of rope that is vormally no movement of the rope. A lanyard, being running rigging, would be moved or adjusted, so blocks or fairleads would be preferred. Standing rigging, which is seldom worked, would be the lines that bullseyes would normally be used for. Bulleyes were frequently used in lieu of deadeyes in standing rigging from the late 18th century on.
Are you sure you meant bullseyes? Running rigging was usually redirected using blocks. Perhaps other members can shed some light on how to manage the web of running rigging such that lines do not get twisted around each other or tangled. I've always run lanyards and other lines through lubbers holes on tops, being careful in arranging the so as not to allow them to be twisted around each other.