  Questions that seek to determine whether a hyperplane arrangement property,
be it geometric, arithmetic or topological, is of a combinatorial nature (that
is determined by the intersection lattice) are abundant in the literature. To
tackle such questions and provide a negative answer, one of the most effective
methods is to produce a counterexample. To this end, it is essential to know
how to construct arrangements that are lattice-equivalent. The more different
they are, the more efficient it will be. In this paper, we present a method to
construct arrangements of complex projective lines that are lattice-equivalent
but lie in distinct connected components of their moduli space. To illustrate
the efficiency of the method, we apply it to reconstruct all the classical
examples of arrangements with disconnected moduli spaces: MacLane,
Falk-Sturmfels, Nasir-Yoshinaga and Rybnikov. Moreover, we employ this method
to produce novel examples of arrangements of eleven lines whose moduli spaces
are formed by four connected components.
