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Following my own advice from my answer to Is there an equivalent to βThe Man in Seat Sixty-Oneβ for bus travel?, I looked on the Lonely Planet page for Tbilisi:
A minifleet of four or five buses run by different companies departs at noon for Istanbul (US$40, 27 hours) via the border at Sarpi, Hopa, Rize
They leave from the second bus terminal:
Ortachala bus station (753433; Gulia 1) is something of a backwater, about 2.5km southeast of the Old Town.