  We use the theory of trianguline $(\varphi,\Gamma)$-modules over pseudorigid
spaces to prove a modularity lifting theorem for certain Galois representations
which are trianguline at $p$, including those with characteristic $p$
coefficients. The use of pseudorigid spaces lets us construct integral models
of the trianguline varieties of [BHS17b], [Che13] after bounding the slope, and
we carry out a Taylor--Wiles patching argument for families of overconvergent
modular forms. This permits us to construct a patched quaternionic eigenvariety
and deduce our modularity results.
