  We initiate the study of Simultaneous Graph Embedding with Fixed Edges in the
beyond planarity framework. In the QuaSEFE problem, we allow edge crossings, as
long as each graph individually is drawn quasiplanar, that is, no three edges
pairwise cross. We show that a triple consisting of two planar graphs and a
tree admit a QuaSEFE. This result also implies that a pair consisting of a
1-planar graph and a planar graph admits a QuaSEFE. We show several other
positive results for triples of planar graphs, in which certain structural
properties for their common subgraphs are fulfilled. For the case in which
simplicity is also required, we give a triple consisting of two quasiplanar
graphs and a star that does not admit a QuaSEFE. Moreover, in contrast to the
planar SEFE problem, we show that it is not always possible to obtain a QuaSEFE
for two matchings if the quasiplanar drawing of one matching is fixed.
