Missing a closing date tends to spur a negative chain reaction in the selling/buying cycle, wreaking havoc on plans, vans and hired hands. Unfortunately, these inconvenient "misses" are all too common, and the vast majority of them are caused by the buyer's side; some agents say as much as 90 percent of the time.
You may well be due some kind of recompense. However, most contracts these days allow a "reasonable" amount of time for performance, although more pointed ones can push punctuality with such things as "time is of the essence" clauses that call for penalties or outright termination if the closing's not met. Check your contract carefully.
If you do have contractual clout, your next move may depend on your degree of motivation. If the closing date is an ironclad term of the contract, you may have a right to kill the deal and keep all or part of the buyer's earnest money, which would hypothetically bail you out of your predicament. But would you really want to do that after going this far with the buyer? Of course, you would lose your deposit on the rent home, unless you want to gamble on selling to another buyer.
Alternatively, perhaps you could ask that the buyer release some of that money directly to you on a nonrefundable basis, or you could ask for a daily charge or "per diem," which is typically figured at one-thirtieth of the seller's monthly payment. Or, you could apply a hybrid of the two formulas. In either case, be sure to modify the sales contract so these dollars will be forfeited to you if the buyer fails to close.
By the way, since the buyer is offshore, it doesn't mean he can't give his proxy to his agent or another party who can close without him being present. While snail mail runs slower to and from an offshore oil rig, he'll still be able to receive, sign and send the needed documents.
Another possible complication is that second appraisal the bank is ordering. Obviously, there is a disparity between what the first appraiser said the house is worth and what the seller is offering (and borrowing). The bank doesn't want to get stuck with a property that's worth less than what it's investing, hence that second appraisal. But realize a lower counteroffer may come from the buyer if the second valuation comes back lower than the first.
As if there already aren't enough complications and delays, it's not unusual for at least a couple of weeks to elapse before an appraisal can be performed and an ensuing mortgage commitment letter can be issued. (I think I hear indistinct cursing coming from your direction.)
Alas, you are in a somewhat delicate spot. You don't want to risk losing your sale in your apparently tenuous financial position, but you don't want to be abused by uncertainty, indecision and delay, only to lose the buyer anyway and possibly your rent home.
If you trust your real estate agent, and I hope you're using a good one in this case, let him or her be your guide in how to handle this. I wish you luck!