"Directivity Error" is something of a loaded term that leads some to regard this as an issue that a manufacturer left in, "an error" rather than the natural consequence of the decisions made in the construction. There are a few reasons why there can be discontinuities through the crossover region. There can be a directivity mismatch where the woofer and tweeter have very different directivities at the crossover point, like a dome tweeter on a flat baffle and an 8" woofer. At the crossover point the tweeter is wide and woofer narrow. Vertically separating the drivers will cause lobes to form in the response where the phases of the two drivers cause cancellation at certain angles. The crossover and filters applied to the drivers can change how they respond to each other and have an effect on the combined directivity. The distance between the drivers has quite a large effect on how they combine at the crossover frequency. As
@voodooless pointed out before, it is quite possible to smooth the DI and early reflections response in the crossover, moving the angle of the lobes from pointing at the early reflection points to somewhere else. The lobes still exist in most cases because the crossover frequency is not low enough for the drivers to sum as one.
There is a good set of graphics made by augerpro at this post over on diyaudio where the CTC distance between the drivers has been changed to reflect a certain numbers of wavelengths at the crossover frequency. You can see how the shape of the directivity changes based on how the lobes form.
https://www.diyaudio.com/community/...ations-with-ideal-drivers.380658/post-7063383
This is something Kimmo Saunisto has suggested for some time now, that increasing the CTC distance to be in the range 1.2 to 1.4 wavelengths at crossover smooths the DI and early reflection response in a beneficial way. There is only anecdotal evidence that it improves sound but when it comes to designing speakers he's usually right.
Genelec here have taken the approach of using high order filters with phase compensation to limit the range of crossover interaction and also the output of the drivers at their frequency extremes. As long as you don't put yourself right in the vertical null, what's left is a small power dip and DI bump, something which there is some old research on from Lipshitz and Vanderkooy.
https://www.aes.org/e-lib/browse.cfm?elib=11454