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We report on the discovery of a new, transient ultraluminous X-ray source (ULX) in the galaxy NGC 7090. This new ULX, which we refer to as NGC 7090 ULX3, was discovered via monitoring with $Swift$ during 2019-20, and to date has exhibited a peak luminosity of $L_{\rm{X}} \sim 6 \times 10^{39}$ erg s$^{-1}$. Archival searches show that, prior to its recent transition into the ULX regime, ULX3 appeared to exhibit a fairly stable luminosity of $L_{\rm{X}} \sim 10^{38}$ erg s$^{-1}$. Such strong long-timescale variability may be reminiscent of the small population of known ULX pulsars, although deep follow-up observations with $XMM$-$Newton$ and $NuSTAR$ do not reveal any robust X-ray pulsation signals. Pulsations similar to those seen from known ULX pulsars cannot be completely excluded, however, as the limit on the pulsed fraction of any signal that remains undetected in these data is $\lesssim$20\%. The broadband spectrum from these observations is well modelled with a simple thin disc model, consistent with sub-Eddington accretion, which may instead imply a moderately large black hole accretor ($M_{\rm{BH}} \sim 40 ~ M_{\odot}$). Similarly, though, more complex models consistent with the super-Eddington spectra seen in other ULXs (and the known ULX pulsars) cannot be excluded given the limited signal-to-noise of the available broadband data. The nature of the accretor powering this new ULX therefore remains uncertain.