In several families of iron-based superconducting materials, a d-wave pairing instability may compete with the leading s-wave instability. Here, we show that when both states have comparable free energies, superconducting and nematic degrees of freedom are strongly coupled. Whereas nematic order causes a sharp nonanalytic increase in Tc, nematic fluctuations can change the character of the s-wave to d-wave transition, favoring an intermediate state that does not break time-reversal symmetry but does break tetragonal symmetry. The coupling between superconductivity and nematicity is also manifested in the strong softening of the shear modulus across the superconducting transition. Our results show that nematicity can be used as a diagnostic tool to search for unconventional pairing states in iron pnictides and chalcogenides.