In this work, iterative modal solvers are proposed for building multiphysics finite element reduced order models. The strongly coupled problems considered are those with both the structure of systems of differential-algebraic equations and sparse discrete operators. Common cases include piezoelectric models and vibroacoustic models with only fluid added-mass effects. The approach is based on the Model Order Reduction (MOR) after Implicit Schur method [1] proposed for the Krylov subspace-based model order reduction of piezoelectric devices. While their work uses the knowledge of the loading applied to the model to generate a specialized reduction basis, it is proposed to build by modal synthesis a reduction basis with a priori unknown loading. The basis is built from the eigenvectors of the problem after the static condensation by Schur complement of one of the physics. Typically, the Schur complement matrix is formed explicitly but it leads to dense operators [2] which limit the problem scales that can be studied due to large memory requirements and costly computations for the eigensolver used afterward. For Krylov-subspace based eigensolvers, the most computationally difficult step is to obtain a basis spanning the eigenspace of the problem on the considered eigenvalue range. By generalizing the MOR after Implicit Schur method, this basis can be constructed by an iterative procedure using the original sparse operators instead of the dense condensed operators. While the original model may be significantly larger compared to the condensed model for typical cases, the conserved sparsity is a critical computational advantage for large scale problems. This method is minimally intrusive for the eigensolvers that only require the implementation of a matrix-vector product. Comparing this implicit Schur complement approach to the explicit Schur complement approach shows large computational cost reductions. It also unveils the problem scale limitations of the explicit approach on a modern HPC node.