  We consider a model of Dirac fermions in $2+1$ dimensions with dynamically
generated, anticommuting SO(3) N\'eel and Z$_2$ Kekul\'e mass terms that
permits sign-free quantum Monte Carlo simulations. The phase diagram is
obtained from finite-size scaling and includes a direct and continuous
transition between the N\'eel and Kekul\'e phases. The fermions remain gapped
across the transition, and our data support an emergent SO(4) symmetry unifying
the two order parameters. While the bare symmetries of our model do not allow
for spinon-carrying Z$_3$ vortices in the Kekul\'e mass, the emergent SO(4)
invariance permits an interpretation of the transition in terms of deconfined
quantum criticality. The phase diagram also features a tricritical point at
which N\'eel, Kekul\'e, and semimetallic phases meet. The present, sign-free
approach can be generalized to a variety of other mass terms and thereby
provides a new framework to study exotic critical phenomena.
