  We adapt Seifert's algorithm for classical knots and links to the setting of
tri-plane diagrams for bridge trisected surfaces in the 4-sphere. Our approach
allows for the construction of a Seifert solid that is described by a Heegaard
diagram. The Seifert solids produced can be assumed to have exteriors that can
be built without 3-handles; in contrast, we give examples of Seifert solids
(not coming from our construction) whose exteriors require arbitrarily many
3-handles. We conclude with two classification results. The first shows that
surfaces admitting doubly-standard shadow diagrams are unknotted. The second
says that a $b$-bridge trisection in which some sector contains at least $b-1$
patches is completely decomposable, thus the corresponding surface is
unknotted. This settles affirmatively a conjecture of the second and fourth
authors.
