  We consider arrangements of tropical hyperplanes where the apices of the
hyperplanes are taken to infinity in certain directions. Such an arrangement
defines a decomposition of Euclidean space where a cell is determined by its
`type' data, analogous to the covectors of an oriented matroid. By work of
Develin-Sturmfels and Fink-Rinc\'{o}n, these `tropical complexes' are dual to
(regular) subdivisions of root polytopes, which in turn are in bijection with
mixed subdivisions of certain generalized permutohedra. Extending previous work
with Joswig-Sanyal, we show how a natural monomial labeling of these complexes
describes polynomial relations (syzygies) among `type ideals' which arise
naturally from the combinatorial data of the arrangement. In particular, we
show that the cotype ideal is Alexander dual to a corresponding initial ideal
of the lattice ideal of the underlying root polytope. This leads to novel ways
of studying algebraic properties of various monomial and toric ideals, as well
as relating them to combinatorial and geometric properties. In particular, our
methods of studying the dimension of the tropical complex leads to new formulas
for homological invariants of toric edge ideals of bipartite graphs, which have
been extensively studied in the commutative algebra community.
