  We introduce the classes of TAP groups, in which various types of algebraic
fibring are detected by the non-vanishing of twisted Alexander polynomials. We
show that finitely presented LERF groups lie in the class $\mathsf{TAP}_1(R)$
for every integral domain $R$, and deduce that algebraic fibring is a profinite
property for such groups. We offer stronger results for algebraic fibring of
products of limit groups, as well as applications to profinite rigidity of
Poincar\'e duality groups in dimension $3$ and RFRS groups.
