  When extending the SM, e.g. with vector-like quarks (VLQ), one inevitably
encounters more than one CP violating complex phases. In this work, we develop
a method useful in reducing the number of complex phases in scenarios with one
and two VLQs. We show that these VLQ-models can be described with just one or
two physical complex phases respectively and present an explicit alternative
parametrization for the quark mixing matrix. Making use of this Reduction of
Complex Phases in the case of the Bento-Branco-Parada (BBP) model, i.e. the
simplest implementation of the Nelson-Barr solution to the strong CP problem,
one also recovers the single phase formulation of BBP model. We comment on how
this specific scenario can be distinguished from the general VLQ case with
weak-basis invariants
