  Unitarity is a fundamental property of any theory required to ensure we work
in a theoretically consistent framework. In comparison with the quark sector,
experimental tests of unitarity for the 3x3 neutrino mixing matrix are
considerably weaker. It must be remembered that the vast majority of our
information on the neutrino mixing angles originates from $\overline{\nu}_e$
and $\nu_\mu$ disappearance experiments, with the assumption of unitarity being
invoked to constrain the remaining elements. New physics can invalidate this
assumption for the 3x3 subset and thus modify our precision measurements. We
perform a reanalysis to see how global knowledge is altered when one refits
oscillation results without assuming unitarity, and present $3 \sigma$ ranges
for allowed $U_\text{PMNS}$ elements consistent with all observed phenomena. We
calculate the bounds on the closure of the six neutrino unitarity triangles,
with the closure of the $\nu_e \nu_\mu$ triangle being constrained to be $\leq$
0.03, while the remaining triangles are significantly less constrained to be
$\leq$ 0.1 - 0.2. Similarly for the row and column normalization, we find their
deviation from unity is constrained to be $\leq$ 0.2 - 0.4, for four out of six
such normalisations, while for the $\nu_\mu$ and $\nu_e$ row normalisation the
deviations are constrained to be $\leq$ 0.07, all at the $3\sigma$ CL. We
emphasise that there is significant room for new low energy physics, especially
in the $\nu_\tau$ sector which very few current experiments constrain directly.
