I feels like the present perfect, but instead of the past participle, it has a to-infinitival. Indeed, it seems equivalent in meaning to have not yet + past participle. But if you take out yet, you get have to, suggesting obligation. Are these related?
The OED treats have to in its entry for have,
"7. a. To possess as a duty or thing to be done. With object and dative inf. expressing what is to be done by the subject.Back to 2c
(This is in origin a particular case of 2b.)
b. Hence to have to do: see
DOv. 33c, d.
c. With infinitive: To be under obligation, to be obliged; to be necessitated to do something. It forms a kind of Future of obligation or duty.
[Cf. the Future tense of the Romanic langs, e.g. je parler-ai, je finir-ai, I have to speak, to finish.]"
" b. with complement or adverbial extension, particularizing the relation of the object or expressing some qualification, condition or limitation thereof.And following up the link to do, section 33:
c1000Ags. Gosp. Matt. iii. 9 We habbaabraham us to fæder. Ibid. John viii. 41 We habbaanne god to fæder. c1290Beket2042 in S. Eng. Leg. I. 165 eto ekingus wille is bodi ehabben al-are. a1300Cursor M. 15317 He athas his bodi clene. 1388WYCLIF 1 Tim. iv. 2 That..haue her conscience corrupt. 1474CAXTON Chesse II. iv. Civb, A knyght which had to name malechete. 1526TINDALE Matt. iii. 4 This Jhon had his garment off camels heer. Ibid. xxii. 11 A man which had not on a weddinge garment. 1583HOLLYBAND Campo di Fior 183 As long as we have this monkey to our cooke. 1594SHAKES. Rich. III,II. i. 112 When Oxford had me downe, he rescued me. 1634SIR T. HERBERT Trav. 3 They used to have their Wives in common. 1700S. L. tr. Fryke's Voy. E. Ind. 14 We still had France on the left of us. 1807ROBINSON Archæol. Græca I. ii. 21 A person who had a foreigner to his mother. 1847MARRYAT Childr. N. Forest v, You..have the laugh on your side now. 1852THACKERAYEsmond I. iii, They had him to dine with them at the inn. 1891MRS. NEWMAN Begun in Jest I. 112, I have women at work for me."
" c. to have to do, to have something to do, to have business, or concern. what has he to do? what business has he...? arch. and dial...
d. to have to do with (in ME. also to do of, at do with): to have dealings or business with; to have connexion or intercourse (of any kind) with; to have relation to."So, what about have + past participle for the perfect aspect?
" II. As an auxiliary verb. As in the other Germanic (and Romanic) languages, the various moods and tenses of haveare used with the pa. pple. of another verb, to form a series of compound or ‘perfect’ tenses of the latter, expressing action already finished at the time indicated, and answering to the Latin perfect tenses dedi, dederam, dedero,dedisse, etc.
This use arose directly from sense 2b, the object possessed having in agreement with it a passive participle of a transitive verb as attribute or complement; thus, I have my work done = ‘I possess or have my work in a done or finished condition’, whence, by inference of antecedent action from result, the actual sense ‘I have done my work’: cf. the series ‘have you the article ready?’, ‘have you the article completed?’, ‘have you completed the article?’ In some dialects the distinction between the original and developed forms, e.g. ‘He has the house built’, ‘he has built the house’, is still in regular use; with some past participles, as begun, completed, done,finished, etc., it is recognized generally. With transitive verbs the developed use was already frequent in OE.; the pa. pple., which originally agreed in number and case with the object, was sometimes left uninflected. In early ME. the usage is found with verbs of action without an object, whence it was extended to intransitive verbs, especially, at an early date, to the verb to be (as in French and other Romanic languages, and in opposition to continental Teutonic use), as he has been, had been, will have been, etc. (cf. F. il a été, Ger. er ist gewesen). Verbs of motion and position long retained the earlier use of the auxiliary be; and he is gone is still used to express resulting state, while he has gone expresses action. SeeIf you search for this construction in the OED examples, the earliest attestation seems to be in 1862. A Google books search antedates this to 1731.
The CGEL gives only a brief mention of the form in chapter 8 when discussing yet.
No conclusions really, but lots to think about!