  The role of non-stoquasticity in the field of quantum annealing and adiabatic
quantum computing is an actively debated topic. We study a strongly-frustrated
quasi-one-dimensional quantum Ising model on a two-leg ladder to elucidate how
a first-order phase transition with a topological origin is affected by
interactions of the $\pm XX$-type. Such interactions are sometimes known as
stoquastic (negative sign) and non-stoquastic (positive sign) "catalysts".
Carrying out a symmetry-preserving real-space renormalization group analysis
and extensive density-matrix renormalization group computations, we show that
the phase diagrams obtained by these two methods are in qualitative agreement
with each other and reveal that the first-order quantum phase transition of a
topological nature remains stable against the introduction of both $XX$-type
catalysts. This is the first study of the effects of non-stoquasticity on a
first-order phase transition between topologically distinct phases. Our results
indicate that non-stoquastic catalysts are generally insufficient for removing
topological obstacles in quantum annealing and adiabatic quantum computing.
