I'll give the opposing viewpoint: that this ought to be on a Form W-2, one way or the other.
It really depends on how the stay bonus was handled by the original company. Since it was a part of your employment, though, I think it most likely ought to be reported on Form W-2; better yet it might have been better handled if Employer B had paid you the bonus from funds Employer A paid them, but no matter.
Certainly, if you'd received this bonus the day you separated from Company A, you'd have been owed a W-2, right? As such I'd argue that the IRS would suggest this deferral of said payment should not affect how it is treated from a tax point of view vis-a-vis employment or nonemployment compensation. It's a payment for doing your job, even if it's outside of the normal compensation framework.
Most of the documentation I see online refers to taxes being withheld, see for example this SEC example stay bonus which is for a company being acquired by another company (not identical to your situation, but fairly close) and refers to tax withholdings, though it does not clarify W2 vs 1099.
Other documents refer to this sort of thing as a deferred compensation plan, which is an interesting question; if it is essentially that sort of plan, then it's even more complicated since there's a whole bucket of rules related to that. For example, this article discusses the possibility of a stay bonus being treated as a non-qualified deferred compensation plan if it pays out more than 2.5 months after the company's tax year end in the year that it triggers; however, that assumes the bonus is paid to a still-current employee, I think. It's unclear whether that would apply here or not, since you were paid within that timeframe when considering the conditions being triggered (1 year stay) but outside of that when considering the term of your employment.
All in all, I think you probably will be issued a 1099, because companies like to use 1099 when they at all possibly can (avoiding paying employment taxes, and avoiding having to treat you as an employee in their systems), but there's an argument that you ought to be issued a W-2.