  We present a comprehensive theoretical study of interacting and disordered
topological phases of coupled Kitaev wires, which may support further realistic
applications of Majorana fermions. We develop a variety of analytical,
mathematical and numerical methods for one and two-coupled wires, associated
with a topological marker accessible from real-space correlation functions on
the wire(s). We verify the stability of the topological superconducting phase
and quantify disorder effects close to the quantum phase transitions, e.g.
through two-point correlation functions or using a renormalization group (RG)
analysis of disorder. We show for the first time that the double critical Ising
(DCI) phase -- a fractional Majorana liquid characterized by a pair of half
central charges and topological numbers -- is stabilized by strong interactions
against disorder which respects the inversion symmetry between the wires (ie.
parity conservation on each wire). % This phase may be equally realized with
two quantum Ising spin chains coupled through a spin interaction along $z$
direction. In the presence of an inter-wire hopping term, the DCI phase turns
into a protected topological phase with a bulk gap. We study the localization
physics developing along the critical line for weaker interactions.
