Summary: Any totally positive \((k+m)\times n\) matrix induces a map \(\pi_+\) from the positive Grassmannian \(\mathrm{Gr}_+(k,n)\) to the Grassmannian \(\mathrm{Gr}(k,k+m)\), whose image is the amplituhedron \(\mathcal{A}_{n,k,m}\) and is endowed with a top-degree form called the canonical form \(\mathbf{\Omega }(\mathcal{A}_{n,k,m})\). This construction was introduced by N. Arkani-Hamed and J. Trnka [J. High Energy Phys. 2014, No. 10, Paper No. 030, 33 p. (2014; Zbl 1468.81075)], where they showed that \(\mathbf{\Omega }(\mathcal{A}_{n,k,4})\) encodes scattering amplitudes in \(\mathcal{N}=4\) super Yang-Mills theory. One way to compute \(\mathbf{\Omega }(\mathcal{A}_{n,k,m})\) is to subdivide \(\mathcal{A}_{n,k,m}\) into so-called generalized triangles and sum over their associated canonical forms. Hence, the physical computation of scattering amplitudes is reduced to finding the triangulations of \(\mathcal{A}_{n,k,4}\). However, while triangulations of polytopes are fully captured by their secondary and fiber polytopes [L. J. Billera and B. Sturmfels, Ann. Math. (2) 135, No. 3, 527–549 (1992; Zbl 0762.52003); I. M. Gelfand et al., Discriminants, resultants, and multidimensional determinants. Boston, MA: Birkhäuser (1994; Zbl 0827.14036)], the study of triangulations of objects beyond polytopes is still underdeveloped. In this work, we initiate the geometric study of subdivisions of \(\mathcal{A}_{n,k,m}\) in order to establish the notion of secondary amplituhedron. For this purpose, we first extend the projection \(\pi_+\) to a rational map \(\pi :\mathrm{Gr} (k,n)\dashrightarrow \mathrm{Gr} (k,k+m)\) and provide a concrete birational parametrization of the fibers of \(\pi\). We then use this to explicitly describe a rational top-degree form \(\omega_{n,k,m}\) (with simple poles) on the fibers and compute \(\mathbf{\Omega }(\mathcal{A}_{n,k,m})\) as a summation of certain residues of \(\omega_{n,k,m}\). As main application of our approach, we develop a well-structured notion of secondary amplituhedra for conjugate to polytopes, i.e. when \(n-k-1=m\) (even). We show that, in this case, each fiber of \(\pi\) is parametrized by a projective space and its volume form \(\omega_{n,k,m}\) has only poles on a hyperplane arrangement. Using such linear structures, for amplituhedra which are cyclic polytopes or conjugate to polytopes, we show that the Jeffrey-Kirwan residue computes \(\mathbf{\Omega }(\mathcal{A}_{n,k,m})\) from the fiber volume form \(\omega_{n,k,m}\). In particular, we give conceptual proofs of the statements of L. Ferro et al. [J. Phys. A, Math. Theor. 52, No. 4, Article ID 045201, 25 p. (2019; Zbl 1422.81148)]. Finally, we propose a more general framework of fiber positive geometries and analyze new families of examples such as fiber polytopes and Grassmann polytopes.