  In the gravitational collapse of matter beyond spherical symmetry,
gravitational waves are necessarily present. On the other hand, gravitational
waves can collapse to a black hole even without matter. One might therefore
wonder whether the interaction and competition between the matter fields and
gravitational waves affects critical phenomena at the threshold of black hole
formation. As a toy model for this, we study type II critical collapse with two
matter fields in spherical symmetry, namely a scalar field and a Yang-Mills
field. On their own, both display discrete self-similarity (DSS) in type II
critical collapse, and we can take either one of them as a toy model for
gravitational waves. To our surprise, in numerical time evolutions we find
that, for sufficiently good fine-tuning, the scalar field always dominates on
sufficiently small scales. We explain our results by the conjectured existence
of a "quasi-discretely self-similar" (QSS) solution shared by the two fields,
equal to the known Yang-Mills critical solution at infinitely large scales and
the known scalar field critical solution (the Choptuik solution) at infinitely
small scales, with a gradual transition from one field to the other. This QSS
solution itself has only one unstable mode, and so acts as the critical
solution for any mixture of scalar field and Yang-Mills initial data.
