  We experimentally investigate the first-order correlation function of a
trapped Fermi gas in the two-dimensional BEC-BCS crossover. We observe a
transition to a low-temperature superfluid phase with algebraically decaying
correlations. We show that the spatial coherence of the entire trapped system
can be characterized by a single temperature-dependent exponent. We find the
exponent at the transition to be constant over a wide range of interaction
strengths across the crossover. This suggests that the phase transitions in
both the bosonic regime and the strongly interacting crossover regime are of
Berezinskii-Kosterlitz-Thouless-type and lie within the same universality
class. On the bosonic side of the crossover, our data are well-described by
Quantum Monte Carlo calculations for a Bose gas. In contrast, in the strongly
interacting regime, we observe a superfluid phase which is significantly
influenced by the fermionic nature of the constituent particles.
