The rearrangement of a material's electron density during laser irradiation leads to modified nonthermal forces on the atoms that may lead to coherent atomic motions and structural phase transformation on very short time scales. We present ab initio molecular dynamics simulations of a martensitic solid-solid phase transformation in tungsten under conditions of strong electronic excitation. The transformation is ultrafast, taking just over a picosecond, and follows the tetragonal Bain path. To examine whether a solid-solid bcc-fcc phase transformation could occur during laser irradiation, we use two-temperature molecular dynamics (2T-MD) simulations with a specially developed potential dependent on the electronic temperature. Our simulations show that the occurrence of the solid-solid phase transformation is in competition with ultrafast nonthermally assisted melting with the strength of the electron-phonon coupling determining the lifetime of the new solid phase. In tungsten the melting transition is predicted to occur too rapidly for the fcc phase to be detectable during laser irradiation.