Though necessary and advantageous in many fields, the high coherence of lasers is detrimental to their performance in certain applications, including illumination, imaging, and projection. This is due to the formation of coherence artifacts, commonly known as speckles, resulting from the interference of randomly scattering spatially coherent photons. It is possible to resolve this issue by increasing the number of mutually incoherent modes emitted from the laser. In vertical-cavity surface-emitting lasers (VCSELs), this can be performed by designing them to have chaotic cavities. This paves the way toward their use in simultaneous illumination and communication scenarios. Herein, we show that chaotic-cavity broad-area VCSELs can achieve significantly broader modulation bandwidths (up to 5 GHz) and higher data rates (up to 12.6 GB/s) compared to other low-coherence light sources, with a lower speckle contrast. We further report a novel technique for lowering the speckle contrast by carefully designing the AC signal used for communication. We show that the apparent spatial coherence is dramatically decreased by inserting a short chirp signal between symbols. Using this method with a chaotic-cavity VCSEL, the number of apparent modes can be up to 450, compared to 88 modes measured from a conventional broad-area VCSEL (a fivefold increase). In light of the recent advances in visible-light VCSELs, this work shows the potential of low-coherence surface-emitting lasers (LCSELs) in simultaneous illumination and optical wireless communication systems since they combine the high speed of lasers with the excellent illumination properties of light-emitting diodes.