We construct a sequence of pairs of 3–manifolds (M1n, M2n)
each with incompressible torus boundary and with the following two properties:
(1) For Mn the result of a carefully chosen glueing of M1n and
M2n along their boundary tori, the genera (g1n, g2n) of
(M1n, M2n) and the genus gn of Mn satisfy the inequality
gn/(g1n + g2n) < 1/2.
(2) The result of amalgamating certain unstabilized Heegaard
splittings of M1n and M2n to form a Heegaard splitting of
M produces a stabilized Heegaard splitting that can be
destabilized successively n times.