Look for the CRI rating of bulbs that you buy. It's a measurement of how close to a blackbody spectrum the bulb is putting out, the highest fidelity being 100. Note that this is not the temperature measurement, and you can have e.g. 2700K or 5000K bulbs with high CRI.
Newer LED phosphors are typically 90+ CRI, and I commonly find 93 CRI bulbs available off the shelf.
Interesting. The Wikipedia entry mentions SPD and I think that is where I think LEDs fall down—having a skewed and/or incomplete spectrum. Even though it may make certain target colors look correct.
Sunlight diverges significantly from a black-body spectrum because the atmosphere absorbs so many wavelengths.