  The class of solvable many-body problems "of goldfish type" is extended by
including (the additional presence of) three-body forces. The solvable $N$-body
problems thereby identified are characterized by Newtonian equations of motion
featuring 19 arbitrary "coupling constants". Restrictions on these constants
are identified which cause these systems - or appropriate variants of them - to
be isochronous or asymptotically isochronous, i.e. all their solutions to be
periodic with a fixed period (independent of the initial data) or to have this
property up to contributions vanishing exponentially as $t\rightarrow\infty$.
