Matching is a central problem in theoretical computer science, with a large body of work spanning the last five decades. However, understanding matching in the time-space bounded setting remains a longstanding open question, even in the presence of additional resources such as randomness or non-determinism. In this work we study space-bounded machines with access to catalytic space, which is additional working memory that is full with arbitrary data that must be preserved at the end of its computation. Despite this heavy restriction, many recent works have shown the power of catalytic space, its utility in designing classical space-bounded algorithms, and surprising connections between catalytic computation and derandomization. Our main result is that bipartite maximum matching ($MATCH$) can be computed in catalytic logspace ($CL$) with a polynomial time bound ($CLP$). Moreover, we show that $MATCH$ can be reduced to the lossy coding problem for $NC$ circuits ($LOSSY[NC]$). This has consequences for matching, catalytic space, and derandomization: - Matching: this is the first well studied subclass of $P$ which is known to compute $MATCH$, as well as the first algorithm simultaneously using sublinear free space and polynomial time with any additional resources. - Catalytic space: this is the first new problem shown to be in $CL$ since the model was defined, and one which is extremely central and well-studied. - Derandomization: we give the first class $\mathcal{C}$ beyond $L$ for which we exhibit a natural problem in $LOSSY[\mathcal{C}]$ which is not known to be in $\mathcal{C}$, as well as a full derandomization of the isolation lemma in $CL$ in the context of $MATCH$. Our proof combines a number of strengthened ideas from isolation-based algorithms for matching alongside the compress-or-random framework in catalytic computation.