Dual or Multi-Nationality
Multi-nationalities can arise form the different nationalities of parents, and by where one is born - in some cases where ones grandparents were born. Some countries do not address multi-nationality at all, and others specifically prohibit it. Many countries bestow automatic citizenship on a child based on the father's citizenship. Other countries allow the citizenship to be acquired from either or both parents. In some countries, a child is allowed dual citizenship, but require the child to choose only one at age eighteen. Other countries have no restrictions. Only in the US is nationality bestowed by birth on US soil. However, exceptions occur for those on diplomatic status.
US Recognition of Multi-Nationality
The US does not prohibit an individual US citizen from holding dual or multi-nationalities. The oath for US naturalized citizens does not call for the renunciation of any previous nationality. The only requirement the US imposes is that US citizenship is not directly or indirectly renounced by the individual. Direct renunciation requires a specific statement renouncing US citizenship. Indirect renouncement can occur by subsequently serving in the armed forces of a foreign nation, or as a member of a foreign government. The US government requires all US citizens to enter the US on their US passport. The US is unique in that there is no passport control process on leaving the US.
Loss of Philippine Citizenship
Once the US oath of allegiance has been taken by a Filipino at a US Oath Taking Ceremony, the Philippine government interprets this as a an automatic loss of Philippine citizenship. However, the US government makes no formal or informal notice to the Philippine government, who have no direct way of knowing whether any Filipino has in fact taken on another nationality.
The issue would only arise if, on entering the Philippines, an immigration officer at passport control might question why on a direct flight from the US, no US entry stamp appeared in a Filipino passport. There would never be such a stamp for a dual US-Filipino citizen. The issue can be avoided by traveling through intermediate countries, such as Hong Kong, and going through their passport control just to acquire some recent entry and exit stamps.
Re-acquiring Philippine Citizenship
A more direct solution is simply to apply for repatriation under the Philippine Retention and Re-Acquisition Act of 2003. This act merely requires the applicant to swear allegiance to the Philippines, and does not require the renunciation of US citizenship. The applicant must be a natural-born Filipino, meaning one or both parents were Philippine citizens at birth of the applicant.
The drawback is that personal appearance before a Philippine consular officer is required, and the whole process can take several months. Unless one is going to be returning to the Philippines to live and/or buy land or property, there really isn't a big inducement for the trouble required.