  The two-dimensional electron gas at the crystalline LaAlO$_{3}$/SrTiO$_{3}$
(c-LAO/STO) interface has sparked large interest due to its exotic properties
including an intriguing gate-tunable superconducting phase. While there is
growing evidence of pronounced spatial inhomogeneity in the conductivity at
STO-based interfaces, the consequences for superconductivity remain largely
unknown. We study interfaces based on amorphous LAO top layers grown at room
temperature (a-LAO/STO) and demonstrate a superconducting phase similar to
c-LAO/STO, however, with a gate-tunable critical temperature of $460 \,
\mathrm{mK}$, higher than any previously reported values for c-LAO/STO. The
dependence of the superconducting critical current on temperature, magnetic
field and backgate-controlled doping is found to be consistently described by a
model of a random array of Josephson-coupled superconducting domains.
