Short answer. The court grants letters testamentary, or letters of administration with the will annexed, covering all of the testator's estate in the Philippines. After debts and administration expenses are paid, that estate is disposed of according to the will, with any residue handled as the law provides.
What the law says
When a will is thus allowed, the court shall grant letters testamentary, or letters of administration with the will annexed, and such letters testamentary or of administration, shall extend to all the estate of the testator in the Philippines. Such estate, after the payment of just debts and expenses of administration, shall be disposed of according to such will, so far as such will may operate upon it; and the residue, if any, shall be disposed of as is provided by law in cases of estates in the Philippines belonging to persons who are inhabitants of another state or country.
Rule 77, Section 4 — Estate, how administered. Read the full provision →
Letters extend to the Philippine estate only
Once the foreign will is allowed here, the court grants letters testamentary, or letters of administration with the will annexed, and those letters reach all of the testator's estate located in the Philippines, giving the estate here its own local administration even though full probate already happened abroad in the testator's home jurisdiction. This local administration is what actually lets an administrator or executor deal with Philippine assets, since the foreign probate proceeding alone has no direct operative effect here.
The will governs, so far as it can operate
After the just debts and expenses of administration are paid, the Philippine estate is disposed of according to the will itself, to the extent the will's provisions can actually operate on property located here, respecting the testator's dispositions rather than defaulting straight to the ordinary rules on intestate distribution among heirs. This keeps the outcome consistent with what the testator actually wanted, even though the estate is being wound up under a separate, local proceeding.
What happens to any residue
Whatever residue remains after the will has been given effect is not left in limbo. It is disposed of as the law provides for estates in the Philippines belonging to persons who were inhabitants of another state or country, tying the outcome back to the general rules for foreign decedents' Philippine property, rather than leaving a gap simply because the will did not address every contingency.
Why a separate Philippine proceeding is needed at all
Probate abroad settles the will's validity in the jurisdiction where that proceeding was held, but it has no automatic operative effect on property located in the Philippines. Rule 77 bridges that gap by letting a Philippine court allow the already-probated will and issue local letters, so an executor or administrator has actual authority to deal with Philippine assets without having to relitigate the will's validity from scratch in a full Philippine probate proceeding.