  We construct, for the first time, an off-shell supersymmetric continuous spin
gauge theory in 4-dimensional Minkowski spacetime, in both constrained and
unconstrained Lagrangian formulations. As an extension to the on-shell
description [1], we employ an auxiliary field to close the algebra of
supersymmetry transformations off-shell. The 4d $\mathcal{N}=1$ massless
continuous spin supermultiplet is then denoted by $(\mathrm{\Phi}, \mathrm{H}
\,; \mathrm{\Psi})$, comprised of a dynamical and a non-dynamical complex
scalar continuous spin gauge fields $\mathrm{\Phi}$ and $\mathrm{H}$, as well
as a Dirac continuous spin gauge field $\mathrm{\Psi}$. In particular, we
demonstrate that the off-shell continuous spin supermultiplet, in a limit,
reproduces off-shell supersymmetry transformations of the known scalar
supermultiplet $(\,{\scriptstyle 0}\,,\, {\scriptstyle 1/2}\,)$, all
integer-spin supermultiplets $(\,s\,,\, s\, {\scriptstyle +\, 1/2}\,),\,
s\geqslant1$, and all half-integer spin supermultiplets $(\, s\, {\scriptstyle
-\, 1/2}\,,\, s \,\,),\, s\geqslant1$.
