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The symmetric orbifold of $\mathbb{T}^4$ was recently shown to be exactlydual to string theory on ${\rm AdS}_3\times {\rm S}^3 \times \mathbb{T}^4$ withminimal ($k=1$) NS-NS flux. The worldsheet theory is best formulated in termsof the hybrid formalism of Berkovits, Vafa & Witten (BVW), in terms of whichthe ${\rm AdS}_3\times {\rm S}^3$ factor is described by a$\mathfrak{psu}(1,1|2)_k$ WZW model. At level $k=1$, $\mathfrak{psu}(1,1|2)_1$has a free field realisation that is obtained from that of$\mathfrak{u}(1,1|2)_1$ upon setting a $\mathfrak{u}(1)$ field, often called$Z$, to zero. We show that the free field version of the ${\cal N}=2$generators of BVW (whose cohomology defines the physical states) does not giverise to an ${\cal N}=2$ algebra, but is rather contaminated by termsproportional to the $Z$-field. We also show how to overcome this problem byintroducing additional ghost fields that implement the quotienting by $Z$.

The free field realisation of the BVW string